Properties

Label 2-322-161.61-c1-0-6
Degree $2$
Conductor $322$
Sign $0.984 + 0.173i$
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  + (−0.786 − 0.618i)2-s + (0.325 + 0.232i)3-s + (0.235 + 0.971i)4-s + (3.61 − 0.697i)5-s + (−0.112 − 0.383i)6-s + (−0.0131 + 2.64i)7-s + (0.415 − 0.909i)8-s + (−0.928 − 2.68i)9-s + (−3.27 − 1.68i)10-s + (0.571 + 0.727i)11-s + (−0.148 + 0.371i)12-s + (2.25 + 3.50i)13-s + (1.64 − 2.07i)14-s + (1.34 + 0.612i)15-s + (−0.888 + 0.458i)16-s + (−0.296 − 0.282i)17-s + ⋯
L(s)  = 1  + (−0.555 − 0.437i)2-s + (0.188 + 0.133i)3-s + (0.117 + 0.485i)4-s + (1.61 − 0.311i)5-s + (−0.0460 − 0.156i)6-s + (−0.00498 + 0.999i)7-s + (0.146 − 0.321i)8-s + (−0.309 − 0.894i)9-s + (−1.03 − 0.533i)10-s + (0.172 + 0.219i)11-s + (−0.0429 + 0.107i)12-s + (0.624 + 0.972i)13-s + (0.439 − 0.553i)14-s + (0.346 + 0.158i)15-s + (−0.222 + 0.114i)16-s + (−0.0719 − 0.0686i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36001 - 0.119209i\)
\(L(\frac12)\) \(\approx\) \(1.36001 - 0.119209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.786 + 0.618i)T \)
7 \( 1 + (0.0131 - 2.64i)T \)
23 \( 1 + (-1.26 + 4.62i)T \)
good3 \( 1 + (-0.325 - 0.232i)T + (0.981 + 2.83i)T^{2} \)
5 \( 1 + (-3.61 + 0.697i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-0.571 - 0.727i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (-2.25 - 3.50i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.296 + 0.282i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (3.80 - 3.62i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-2.33 + 0.686i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.473 + 4.95i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-9.67 + 3.34i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-1.72 + 1.49i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (8.39 - 3.83i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-5.29 - 3.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (13.0 + 0.622i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (4.95 - 9.61i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (4.49 + 6.31i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (2.16 + 5.41i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (0.326 - 2.26i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (7.85 - 1.90i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (11.2 - 0.535i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-7.75 + 8.94i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-6.06 - 0.579i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (4.41 + 5.08i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.58838857689651620657227426858, −10.47630560380032713558305580224, −9.372552043085804817997750970577, −9.221601234981627126800624824236, −8.270055068706820722183918327631, −6.37207377475725411741833820196, −6.04003538310606285426624077873, −4.40681871355958237194940523166, −2.72486373176074591256139919255, −1.67733977897380675055705093693, 1.48099752693257033203762315506, 2.92469859791028568244814312281, 4.91855712302305293800095384370, 5.95807148106324311878839301134, 6.78990480712615208057788863252, 7.87284015278599577284468307365, 8.826837429154779228132617196080, 9.839914767066646793907147476568, 10.59707925154899880410960316709, 11.09525783692816393874068616756

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