Properties

Label 4-18e4-1.1-c2e2-0-14
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $77.9399$
Root an. cond. $2.97125$
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  + 4·2-s + 12·4-s + 2·5-s + 12·7-s + 32·8-s + 8·10-s − 12·11-s − 2·13-s + 48·14-s + 80·16-s + 20·17-s + 24·20-s − 48·22-s − 48·23-s + 25·25-s − 8·26-s + 144·28-s + 26·29-s + 12·31-s + 192·32-s + 80·34-s + 24·35-s + 52·37-s + 64·40-s − 58·41-s + 84·43-s − 144·44-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 2/5·5-s + 12/7·7-s + 4·8-s + 4/5·10-s − 1.09·11-s − 0.153·13-s + 24/7·14-s + 5·16-s + 1.17·17-s + 6/5·20-s − 2.18·22-s − 2.08·23-s + 25-s − 0.307·26-s + 36/7·28-s + 0.896·29-s + 0.387·31-s + 6·32-s + 2.35·34-s + 0.685·35-s + 1.40·37-s + 8/5·40-s − 1.41·41-s + 1.95·43-s − 3.27·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(77.9399\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(11.73622581\)
\(L(\frac12)\) \(\approx\) \(11.73622581\)
\(L(2)\) 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$C_1$ \( ( 1 - p T )^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 12 T + 169 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2^2$ \( 1 + 2 T - 165 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 48 T + 1297 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 26 T - 165 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 12 T + 1009 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 58 T + 1683 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2^2$ \( 1 - 84 T + 4201 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 120 T + 7009 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 156 T + 11593 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 26 T - 3045 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 12 T + 4537 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 204 T + 20113 T^{2} + 204 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 167 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \)
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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

−11.70175612991605185008161303429, −11.38450157546239039523420068052, −10.72708856817599293653862684837, −10.67183922304515160595772983314, −9.930262170102490234114113184780, −9.672432328679105085508528927590, −8.251667108022740096739230768903, −8.221324849462244508245156427103, −7.64291755911515966455585913226, −7.42232202005322908874275671669, −6.31726624825752320991366781181, −6.20722678184229780528759230076, −5.53234246021590886110051399242, −5.04791950917796816308725988536, −4.50308972591366444918788554264, −4.40030990816663310216133463887, −3.06211886144995575029150057950, −2.96266961726695260431721378248, −1.83627215909546123013354652938, −1.49738287502713629169316420775, 1.49738287502713629169316420775, 1.83627215909546123013354652938, 2.96266961726695260431721378248, 3.06211886144995575029150057950, 4.40030990816663310216133463887, 4.50308972591366444918788554264, 5.04791950917796816308725988536, 5.53234246021590886110051399242, 6.20722678184229780528759230076, 6.31726624825752320991366781181, 7.42232202005322908874275671669, 7.64291755911515966455585913226, 8.221324849462244508245156427103, 8.251667108022740096739230768903, 9.672432328679105085508528927590, 9.930262170102490234114113184780, 10.67183922304515160595772983314, 10.72708856817599293653862684837, 11.38450157546239039523420068052, 11.70175612991605185008161303429

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