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Properties

Label 2-322-161.103-c1-0-14
Degree $2$
Conductor $322$
Sign $-0.959 + 0.281i$
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  + (−0.0475 − 0.998i)2-s + (−0.673 − 1.68i)3-s + (−0.995 + 0.0950i)4-s + (0.997 − 4.11i)5-s + (−1.64 + 0.752i)6-s + (2.64 − 0.0740i)7-s + (0.142 + 0.989i)8-s + (−0.206 + 0.196i)9-s + (−4.15 − 0.800i)10-s + (−1.93 − 0.0922i)11-s + (0.830 + 1.61i)12-s + (2.09 + 1.81i)13-s + (−0.199 − 2.63i)14-s + (−7.58 + 1.09i)15-s + (0.981 − 0.189i)16-s + (−0.0427 + 0.0600i)17-s + ⋯
L(s)  = 1  + (−0.0336 − 0.706i)2-s + (−0.388 − 0.971i)3-s + (−0.497 + 0.0475i)4-s + (0.446 − 1.83i)5-s + (−0.673 + 0.307i)6-s + (0.999 − 0.0279i)7-s + (0.0503 + 0.349i)8-s + (−0.0687 + 0.0655i)9-s + (−1.31 − 0.253i)10-s + (−0.583 − 0.0278i)11-s + (0.239 + 0.465i)12-s + (0.581 + 0.504i)13-s + (−0.0533 − 0.705i)14-s + (−1.95 + 0.281i)15-s + (0.245 − 0.0473i)16-s + (−0.0103 + 0.0145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.959 + 0.281i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.959 + 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.179853 - 1.25357i\)
\(L(\frac12)\) \(\approx\) \(0.179853 - 1.25357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0475 + 0.998i)T \)
7 \( 1 + (-2.64 + 0.0740i)T \)
23 \( 1 + (4.67 - 1.06i)T \)
good3 \( 1 + (0.673 + 1.68i)T + (-2.17 + 2.07i)T^{2} \)
5 \( 1 + (-0.997 + 4.11i)T + (-4.44 - 2.29i)T^{2} \)
11 \( 1 + (1.93 + 0.0922i)T + (10.9 + 1.04i)T^{2} \)
13 \( 1 + (-2.09 - 1.81i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (0.0427 - 0.0600i)T + (-5.56 - 16.0i)T^{2} \)
19 \( 1 + (-4.27 - 6.00i)T + (-6.21 + 17.9i)T^{2} \)
29 \( 1 + (-0.634 - 1.38i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-4.70 - 5.98i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (0.969 + 1.01i)T + (-1.76 + 36.9i)T^{2} \)
41 \( 1 + (-0.280 - 0.956i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-11.5 - 1.65i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (5.09 + 2.94i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0302 + 0.0104i)T + (41.6 + 32.7i)T^{2} \)
59 \( 1 + (0.651 - 3.38i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (-1.05 - 0.421i)T + (44.1 + 42.0i)T^{2} \)
67 \( 1 + (-3.10 + 6.02i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (2.49 - 1.60i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (0.918 + 9.62i)T + (-71.6 + 13.8i)T^{2} \)
79 \( 1 + (6.64 - 2.29i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (-12.6 - 3.70i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (13.1 + 10.3i)T + (20.9 + 86.4i)T^{2} \)
97 \( 1 + (1.81 - 0.533i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.63232588160762996546339656188, −10.30266191366258082666011221975, −9.307188443673510477231387893976, −8.335089269421067419024959141919, −7.72742744752123989541359699604, −5.98243091662427820283272741458, −5.18065711864536830997391954976, −4.11941530926477562840755816390, −1.79626075746044182948297896239, −1.12237823391569376753687497338, 2.62373492905087121702843500860, 4.06712458523148869461334272618, 5.25800104318283326166622711815, 6.08886147159020635587866012253, 7.26146297246712220195831056526, 8.006681762693163152666835010984, 9.498815365625660379306202579474, 10.26879466092042191163731018908, 10.93300416322929533199050473517, 11.54410662608784811989745007000

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