Properties

Label 8-136e4-1.1-c2e4-0-0
Degree $8$
Conductor $342102016$
Sign $1$
Analytic cond. $188.580$
Root an. cond. $1.92502$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·3-s + 46·9-s − 52·11-s − 16·16-s − 48·17-s + 48·19-s − 184·27-s + 416·33-s − 96·41-s + 28·43-s + 128·48-s + 384·51-s − 384·57-s + 164·59-s + 336·67-s + 48·73-s + 610·81-s + 316·83-s + 428·97-s − 2.39e3·99-s − 524·107-s − 288·113-s + 1.30e3·121-s + 768·123-s + 127-s − 224·129-s + 131-s + ⋯
L(s)  = 1  − 8/3·3-s + 46/9·9-s − 4.72·11-s − 16-s − 2.82·17-s + 2.52·19-s − 6.81·27-s + 12.6·33-s − 2.34·41-s + 0.651·43-s + 8/3·48-s + 7.52·51-s − 6.73·57-s + 2.77·59-s + 5.01·67-s + 0.657·73-s + 7.53·81-s + 3.80·83-s + 4.41·97-s − 24.1·99-s − 4.89·107-s − 2.54·113-s + 10.7·121-s + 6.24·123-s + 0.00787·127-s − 1.73·129-s + 0.00763·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(188.580\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{136} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 17^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2240365261\)
\(L(\frac12)\) \(\approx\) \(0.2240365261\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + p^{4} T^{4} \)
17$C_2^2$ \( 1 + 48 T + 1152 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \)
good3$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T^{2} + p^{4} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \)
5$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
7$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2}( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
29$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
31$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
37$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
41$C_2^2$$\times$$C_2^2$ \( ( 1 - 1246 T^{2} + p^{4} T^{4} )( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} ) \)
43$C_2$$\times$$C_2^2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2}( 1 - 3502 T^{2} + p^{4} T^{4} ) \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + p^{4} T^{4} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 - 82 T + p^{2} T^{2} )^{2}( 1 - 238 T^{2} + p^{4} T^{4} ) \)
61$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
67$C_2^2$ \( ( 1 - 168 T + 14112 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )( 1 + 9506 T^{2} + p^{4} T^{4} ) \)
79$C_4\times C_2$ \( 1 + p^{8} T^{8} \)
83$C_2$$\times$$C_2^2$ \( ( 1 - 158 T + p^{2} T^{2} )^{2}( 1 + 11186 T^{2} + p^{4} T^{4} ) \)
89$C_2^2$$\times$$C_2^2$ \( ( 1 - 144 T + 10368 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )( 1 + 144 T + 10368 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} ) \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 94 T + p^{2} T^{2} )^{2}( 1 - 240 T + 28800 T^{2} - 240 p^{2} T^{3} + p^{4} T^{4} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.749222850149932454411201964159, −9.318967917415573251398136815639, −9.135317368552426831272483295389, −8.363421763086942462675279237122, −8.311695761603839241481703169737, −8.037503672281955575631037346500, −7.55799808205574953853409759381, −7.45533150013843930075509543139, −7.10181623202252572917931738782, −6.66412501359125649282107588343, −6.54249291934765986263438840273, −6.53044857902942224821520901712, −5.66559296558339253301988464031, −5.34064946866851208897943966489, −5.29388460417709564104902002898, −5.03374573111986217646359261119, −4.82484061771453811918804069193, −4.73050040326917677671917785575, −3.81398568526402610069685642336, −3.69997783574079313346639959300, −2.70382030510955332830316749314, −2.29330489815838369534410705368, −2.18670113939316599113085848528, −0.843286311765060617672073876087, −0.29118608353256874582739739615, 0.29118608353256874582739739615, 0.843286311765060617672073876087, 2.18670113939316599113085848528, 2.29330489815838369534410705368, 2.70382030510955332830316749314, 3.69997783574079313346639959300, 3.81398568526402610069685642336, 4.73050040326917677671917785575, 4.82484061771453811918804069193, 5.03374573111986217646359261119, 5.29388460417709564104902002898, 5.34064946866851208897943966489, 5.66559296558339253301988464031, 6.53044857902942224821520901712, 6.54249291934765986263438840273, 6.66412501359125649282107588343, 7.10181623202252572917931738782, 7.45533150013843930075509543139, 7.55799808205574953853409759381, 8.037503672281955575631037346500, 8.311695761603839241481703169737, 8.363421763086942462675279237122, 9.135317368552426831272483295389, 9.318967917415573251398136815639, 9.749222850149932454411201964159

Graph of the $Z$-function along the critical line