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Properties

Label 2-396-11.3-c1-0-2
Degree $2$
Conductor $396$
Sign $0.970 + 0.242i$
Analytic cond. $3.16207$
Root an. cond. $1.77822$
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  + (2.30 + 1.67i)5-s + (1.30 − 4.02i)7-s + (−2.19 − 2.48i)11-s + (1.42 − 1.03i)13-s + (3.73 + 2.71i)17-s + (1.88 + 5.79i)19-s − 4.23·23-s + (0.972 + 2.99i)25-s + (1.38 − 4.25i)29-s + (6.97 − 5.06i)31-s + (9.78 − 7.10i)35-s + (−2.54 + 7.83i)37-s + (−0.163 − 0.502i)41-s + 0.527·43-s + (−0.427 − 1.31i)47-s + ⋯
L(s)  = 1  + (1.03 + 0.750i)5-s + (0.494 − 1.52i)7-s + (−0.660 − 0.750i)11-s + (0.395 − 0.287i)13-s + (0.906 + 0.658i)17-s + (0.431 + 1.32i)19-s − 0.883·23-s + (0.194 + 0.598i)25-s + (0.256 − 0.789i)29-s + (1.25 − 0.909i)31-s + (1.65 − 1.20i)35-s + (−0.418 + 1.28i)37-s + (−0.0254 − 0.0784i)41-s + 0.0804·43-s + (−0.0622 − 0.191i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65873 - 0.203903i\)
\(L(\frac12)\) \(\approx\) \(1.65873 - 0.203903i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (2.19 + 2.48i)T \)
good5 \( 1 + (-2.30 - 1.67i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.30 + 4.02i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.42 + 1.03i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.73 - 2.71i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.88 - 5.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.97 + 5.06i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.54 - 7.83i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.163 + 0.502i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.527T + 43T^{2} \)
47 \( 1 + (0.427 + 1.31i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.9 - 7.97i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.73 - 8.42i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.309 - 0.224i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.85T + 67T^{2} \)
71 \( 1 + (2.92 + 2.12i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.381 - 1.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.89 - 5.73i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.28 - 3.83i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + (4.92 - 3.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.96525873764137692072442398570, −10.15847929981949144288225950151, −10.02216806020382789641884630141, −8.166447600240460563442606545577, −7.70781291584179034552219271839, −6.33493516452842278553500034223, −5.71772404133654698623412320939, −4.19388352506057082099240223189, −3.04495654349904975591788200225, −1.38607121349315271436017041686, 1.70700392044004889295753684338, 2.79372108428957246267813875803, 4.93198934925697264532986774986, 5.28601675650963109582570121040, 6.38464440367668299828377706414, 7.75413424052055130150511028157, 8.819323724308963383685400334509, 9.340242717797265077139944277363, 10.23767096343231656169433669578, 11.50136815293467024593157370492

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