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Properties

Label 2-322-7.4-c1-0-3
Degree $2$
Conductor $322$
Sign $0.960 - 0.277i$
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 + (0.0985 − 0.170i)3-s + (−0.499 + 0.866i)4-s + (0.154 + 0.267i)5-s − 0.197·6-s + (−1.71 + 2.01i)7-s + 0.999·8-s + (1.48 + 2.56i)9-s + (0.154 − 0.267i)10-s + (−0.184 + 0.320i)11-s + (0.0985 + 0.170i)12-s + 6.29·13-s + (2.60 + 0.471i)14-s + 0.0608·15-s + (−0.5 − 0.866i)16-s + (−2.30 + 3.99i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.0569 − 0.0985i)3-s + (−0.249 + 0.433i)4-s + (0.0690 + 0.119i)5-s − 0.0804·6-s + (−0.646 + 0.762i)7-s + 0.353·8-s + (0.493 + 0.854i)9-s + (0.0488 − 0.0845i)10-s + (−0.0557 + 0.0965i)11-s + (0.0284 + 0.0492i)12-s + 1.74·13-s + (0.695 + 0.126i)14-s + 0.0157·15-s + (−0.125 − 0.216i)16-s + (−0.559 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.960 - 0.277i$
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: \((2,\ 322,\ (\ :1/2),\ 0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06563 + 0.150670i\)
\(L(\frac12)\) \(\approx\) \(1.06563 + 0.150670i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 2.01i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.0985 + 0.170i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.154 - 0.267i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.184 - 0.320i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.29T + 13T^{2} \)
17 \( 1 + (2.30 - 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.176 - 0.305i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 1.74T + 29T^{2} \)
31 \( 1 + (1.46 - 2.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.18 - 5.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 - 6.55T + 43T^{2} \)
47 \( 1 + (5.86 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.95 - 5.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.14 + 1.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.39 + 5.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.37 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.492T + 71T^{2} \)
73 \( 1 + (2.99 - 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.333 + 0.576i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.44T + 83T^{2} \)
89 \( 1 + (5.81 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.96T + 97T^{2} \)
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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

−11.52037162524675570010138417778, −10.68698167565940446474581331070, −9.981313409834506102251417882073, −8.749844388179610188492143088917, −8.273169652071912158831778202102, −6.82281617963874641748641993701, −5.83665439066080360907782063981, −4.34395857203823037259302264529, −3.07104739073207130633112884700, −1.71388420520114867792770960064, 0.972352487748841390901780782432, 3.38667664162145454717527283792, 4.46180931334775298891975094805, 5.99859195142265320215289607619, 6.72102030998595542596417369400, 7.65805689745330542435892236275, 8.972273007633186031041495568040, 9.426343967381397233722733089507, 10.57175120680250293176557999908, 11.32836110623154202007695471275

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