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Properties

Label 2-322-7.4-c1-0-12
Degree $2$
Conductor $322$
Sign $0.0432 + 0.999i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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 + (−1.31 + 2.27i)3-s + (−0.499 + 0.866i)4-s + (−1.68 − 2.91i)5-s − 2.62·6-s + (−1.55 − 2.13i)7-s − 0.999·8-s + (−1.94 − 3.36i)9-s + (1.68 − 2.91i)10-s + (1.11 − 1.93i)11-s + (−1.31 − 2.27i)12-s − 3.10·13-s + (1.07 − 2.41i)14-s + 8.85·15-s + (−0.5 − 0.866i)16-s + (−3.18 + 5.51i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.757 + 1.31i)3-s + (−0.249 + 0.433i)4-s + (−0.753 − 1.30i)5-s − 1.07·6-s + (−0.589 − 0.807i)7-s − 0.353·8-s + (−0.648 − 1.12i)9-s + (0.533 − 0.923i)10-s + (0.335 − 0.581i)11-s + (−0.378 − 0.656i)12-s − 0.861·13-s + (0.286 − 0.646i)14-s + 2.28·15-s + (−0.125 − 0.216i)16-s + (−0.772 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0432 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0432 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.0432 + 0.999i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.0432 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152397 - 0.145942i\)
\(L(\frac12)\) \(\approx\) \(0.152397 - 0.145942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.55 + 2.13i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (1.31 - 2.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.68 + 2.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.11 + 1.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.10T + 13T^{2} \)
17 \( 1 + (3.18 - 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.457 + 0.791i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 4.22T + 29T^{2} \)
31 \( 1 + (-1.73 + 3.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.70 + 8.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.85T + 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + (3.28 + 5.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.57 - 9.65i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.48 + 2.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.43 - 5.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.61T + 71T^{2} \)
73 \( 1 + (2.96 - 5.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.37 - 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.51T + 83T^{2} \)
89 \( 1 + (-0.792 - 1.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−11.35531656546841138949783136355, −10.52174528634585460325258661588, −9.435437183996506176516239417193, −8.697614925506222436550401153590, −7.54908699585792606486299850196, −6.22730840358083795177333246783, −5.22360038434488930780866150529, −4.24250361468833011411746974426, −3.84382252835713366516145792981, −0.14032784609561434340740717116, 2.15771577716793156206957020363, 3.17996774258039127593581343547, 4.89130851864802570996357816946, 6.22781501354075658829665801927, 6.90795716568955638792495282032, 7.61284806309302347412432522999, 9.212947934654729802875372558407, 10.29835934954092157671709187312, 11.41986847943831802504844903289, 11.81634398760448963222517867275

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