Properties

Label 4-28e4-1.1-c1e2-0-27
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $39.1909$
Root an. cond. $2.50205$
Motivic weight $1$
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  − 2·2-s + 2·3-s + 2·4-s + 2·5-s − 4·6-s + 2·9-s − 4·10-s + 2·11-s + 4·12-s + 2·13-s + 4·15-s − 4·16-s + 4·17-s − 4·18-s − 6·19-s + 4·20-s − 4·22-s + 2·25-s − 4·26-s + 6·27-s + 6·29-s − 8·30-s + 16·31-s + 8·32-s + 4·33-s − 8·34-s + 4·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s + 2/3·9-s − 1.26·10-s + 0.603·11-s + 1.15·12-s + 0.554·13-s + 1.03·15-s − 16-s + 0.970·17-s − 0.942·18-s − 1.37·19-s + 0.894·20-s − 0.852·22-s + 2/5·25-s − 0.784·26-s + 1.15·27-s + 1.11·29-s − 1.46·30-s + 2.87·31-s + 1.41·32-s + 0.696·33-s − 1.37·34-s + 2/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(39.1909\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.115012666\)
\(L(\frac12)\) \(\approx\) \(2.115012666\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.07279512277347440603653360628, −10.06761604020585335844015596440, −9.626397640788617305542826310136, −9.250789149024733211205702691372, −8.786475079386473753854064967290, −8.311910899086996047375692876816, −8.096415615550570672290551793665, −8.049885367649855008995564143277, −7.01229426699151381978972629095, −6.79465377773371511331025753353, −6.17356214159771119692994923744, −6.11520949765605706653710339821, −5.02790186580975300644374568269, −4.56354325005448256043251322387, −4.11208490708982089951605820287, −3.29529530030614037990319148976, −2.61499720909422454098810350945, −2.38572718315493662519846274490, −1.38653036214049434598174881070, −1.01481735915910015118084504780, 1.01481735915910015118084504780, 1.38653036214049434598174881070, 2.38572718315493662519846274490, 2.61499720909422454098810350945, 3.29529530030614037990319148976, 4.11208490708982089951605820287, 4.56354325005448256043251322387, 5.02790186580975300644374568269, 6.11520949765605706653710339821, 6.17356214159771119692994923744, 6.79465377773371511331025753353, 7.01229426699151381978972629095, 8.049885367649855008995564143277, 8.096415615550570672290551793665, 8.311910899086996047375692876816, 8.786475079386473753854064967290, 9.250789149024733211205702691372, 9.626397640788617305542826310136, 10.06761604020585335844015596440, 10.07279512277347440603653360628

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