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Properties

Label 2-402-67.37-c1-0-6
Degree $2$
Conductor $402$
Sign $0.961 - 0.275i$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
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  + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s − 0.760·5-s + (−0.5 − 0.866i)6-s + (1.97 − 3.41i)7-s − 0.999·8-s + 9-s + (−0.380 − 0.658i)10-s + (3.06 − 5.30i)11-s + (0.499 − 0.866i)12-s + (3.30 + 5.71i)13-s + 3.94·14-s + 0.760·15-s + (−0.5 − 0.866i)16-s + (−0.380 − 0.658i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s − 0.340·5-s + (−0.204 − 0.353i)6-s + (0.745 − 1.29i)7-s − 0.353·8-s + 0.333·9-s + (−0.120 − 0.208i)10-s + (0.923 − 1.59i)11-s + (0.144 − 0.249i)12-s + (0.915 + 1.58i)13-s + 1.05·14-s + 0.196·15-s + (−0.125 − 0.216i)16-s + (−0.0922 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{402} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42705 + 0.200233i\)
\(L(\frac12)\) \(\approx\) \(1.42705 + 0.200233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
67 \( 1 + (-1.42 - 8.05i)T \)
good5 \( 1 + 0.760T + 5T^{2} \)
7 \( 1 + (-1.97 + 3.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.06 + 5.30i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.30 - 5.71i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.380 + 0.658i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 5.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.619 - 1.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.68 + 6.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.35 + 2.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.71 + 2.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.978 + 1.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 9.06T + 43T^{2} \)
47 \( 1 + (-1.83 + 3.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.46T + 53T^{2} \)
59 \( 1 + 7.16T + 59T^{2} \)
61 \( 1 + (2.80 + 4.85i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (7.97 - 13.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.921 - 1.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.27 - 5.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.23 + 9.06i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.12T + 89T^{2} \)
97 \( 1 + (-7.05 - 12.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.56017790364146941508042104170, −10.66732614646207176758583288642, −9.363519887967560851784064161345, −8.317085213367720652371770957545, −7.48037212485482799872905854764, −6.46885002416658944412487940943, −5.73332528139423693928848205175, −4.14945965742034999826672857349, −3.88904663941081309364957356259, −1.15693986665128062616549003065, 1.46690200459788896850576065184, 2.95016211163298535249557095455, 4.45509830092782081498442777594, 5.23256430992998182218853129486, 6.21663370419907317812554968886, 7.48564935473951245586777696145, 8.651288654109020576265472358150, 9.497408099338550408667485546691, 10.61385987176135619866270686621, 11.32263197665568054576315906459

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