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Properties

Label 2-322-161.100-c1-0-11
Degree $2$
Conductor $322$
Sign $0.972 - 0.234i$
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 + (1.00 − 1.40i)3-s + (0.235 + 0.971i)4-s + (1.51 − 0.292i)5-s + (1.65 − 0.487i)6-s + (−0.430 + 2.61i)7-s + (−0.415 + 0.909i)8-s + (0.00389 + 0.0112i)9-s + (1.37 + 0.708i)10-s + (0.138 − 0.108i)11-s + (1.60 + 0.642i)12-s + (1.09 − 0.706i)13-s + (−1.95 + 1.78i)14-s + (1.10 − 2.42i)15-s + (−0.888 + 0.458i)16-s + (−4.12 − 3.92i)17-s + ⋯
L(s)  = 1  + (0.555 + 0.437i)2-s + (0.578 − 0.812i)3-s + (0.117 + 0.485i)4-s + (0.678 − 0.130i)5-s + (0.677 − 0.198i)6-s + (−0.162 + 0.986i)7-s + (−0.146 + 0.321i)8-s + (0.00129 + 0.00375i)9-s + (0.434 + 0.223i)10-s + (0.0416 − 0.0327i)11-s + (0.463 + 0.185i)12-s + (0.304 − 0.195i)13-s + (−0.521 + 0.477i)14-s + (0.286 − 0.627i)15-s + (−0.222 + 0.114i)16-s + (−0.999 − 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21609 + 0.263765i\)
\(L(\frac12)\) \(\approx\) \(2.21609 + 0.263765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.430 - 2.61i)T \)
23 \( 1 + (-0.158 + 4.79i)T \)
good3 \( 1 + (-1.00 + 1.40i)T + (-0.981 - 2.83i)T^{2} \)
5 \( 1 + (-1.51 + 0.292i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (-0.138 + 0.108i)T + (2.59 - 10.6i)T^{2} \)
13 \( 1 + (-1.09 + 0.706i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (4.12 + 3.92i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-4.19 + 3.99i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (7.53 - 2.21i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (6.47 - 0.617i)T + (30.4 - 5.86i)T^{2} \)
37 \( 1 + (-0.938 - 2.71i)T + (-29.0 + 22.8i)T^{2} \)
41 \( 1 + (-3.48 - 4.02i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (1.77 + 3.87i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + (1.31 - 2.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.319 - 6.70i)T + (-52.7 - 5.03i)T^{2} \)
59 \( 1 + (8.98 + 4.63i)T + (34.2 + 48.0i)T^{2} \)
61 \( 1 + (-7.56 - 10.6i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (0.973 - 0.389i)T + (48.4 - 46.2i)T^{2} \)
71 \( 1 + (-0.925 + 6.43i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-1.83 - 7.58i)T + (-64.8 + 33.4i)T^{2} \)
79 \( 1 + (0.511 + 10.7i)T + (-78.6 + 7.50i)T^{2} \)
83 \( 1 + (-9.03 + 10.4i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-7.09 - 0.677i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-1.44 - 1.66i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.91641782804219763609658283417, −10.96151017694848044707375008498, −9.357218441400206812049064639039, −8.849181888087159474882621774386, −7.67150914538056593960851032316, −6.82817129636471357107686377198, −5.78442252168236501718322528779, −4.85078013342283302918319600910, −3.01068500708259464388677661345, −2.03858329771635126516351479829, 1.83055730624956805004430011666, 3.57653304429042463253198541318, 4.01920554196165446174232113891, 5.48812907874416874603150180041, 6.54144983575064194515886661616, 7.79736903976231368132374238552, 9.265480694587827738605221319744, 9.739220793556945291528310776572, 10.61255221111334624829581529526, 11.38190158853480219261177791021

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