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Properties

Label 2-1152-16.13-c1-0-12
Degree $2$
Conductor $1152$
Sign $0.962 + 0.270i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.74 − 1.74i)5-s + 2.55i·7-s + (0.473 − 0.473i)11-s + (−2.88 − 2.88i)13-s + 6.44·17-s + (4.55 + 4.55i)19-s − 2.82i·23-s − 1.11i·25-s + (−3.07 − 3.07i)29-s + 6.55·31-s + (4.47 + 4.47i)35-s + (2.72 − 2.72i)37-s − 0.788i·41-s + (0.389 − 0.389i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (0.782 − 0.782i)5-s + 0.966i·7-s + (0.142 − 0.142i)11-s + (−0.800 − 0.800i)13-s + 1.56·17-s + (1.04 + 1.04i)19-s − 0.589i·23-s − 0.223i·25-s + (−0.571 − 0.571i)29-s + 1.17·31-s + (0.756 + 0.756i)35-s + (0.448 − 0.448i)37-s − 0.123i·41-s + (0.0594 − 0.0594i)43-s − 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.956173302\)
\(L(\frac12)\) \(\approx\) \(1.956173302\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1.74 + 1.74i)T - 5iT^{2} \)
7 \( 1 - 2.55iT - 7T^{2} \)
11 \( 1 + (-0.473 + 0.473i)T - 11iT^{2} \)
13 \( 1 + (2.88 + 2.88i)T + 13iT^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + (-4.55 - 4.55i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (3.07 + 3.07i)T + 29iT^{2} \)
31 \( 1 - 6.55T + 31T^{2} \)
37 \( 1 + (-2.72 + 2.72i)T - 37iT^{2} \)
41 \( 1 + 0.788iT - 41T^{2} \)
43 \( 1 + (-0.389 + 0.389i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (2.57 - 2.57i)T - 53iT^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 + (-4.38 - 4.38i)T + 61iT^{2} \)
67 \( 1 + (-2.11 - 2.11i)T + 67iT^{2} \)
71 \( 1 - 5.11iT - 71T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + (-0.641 - 0.641i)T + 83iT^{2} \)
89 \( 1 + 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.838280131469858866310239845481, −9.030025656069348557162681757354, −8.151246799812984017778424450523, −7.49064505479072828174747790040, −6.00522583679409293279697898483, −5.60208039458041338039176108334, −4.87082459901987955939527537793, −3.41431641680874553476076513516, −2.36463404681112542349452444571, −1.08849537860410229588779211743, 1.20334648314940526250947193952, 2.57704024963537383648417071792, 3.52413343893035994883828299635, 4.70469277005981514218970748721, 5.60377034751894040044436323111, 6.71347505666456334929395607106, 7.18393441412354150277593604430, 7.997505066813947680359242835648, 9.394740891726390172170089924915, 9.819928376249494076808812433529

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