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Properties

Label 2-322-161.45-c1-0-10
Degree $2$
Conductor $322$
Sign $0.164 + 0.986i$
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 + (−2.64 − 1.52i)3-s + (−0.499 + 0.866i)4-s + (0.920 + 1.59i)5-s − 3.05i·6-s + (0.254 − 2.63i)7-s − 0.999·8-s + (3.18 + 5.51i)9-s + (−0.920 + 1.59i)10-s + (−3.87 − 2.23i)11-s + (2.64 − 1.52i)12-s − 2.97i·13-s + (2.40 − 1.09i)14-s − 5.63i·15-s + (−0.5 − 0.866i)16-s + (3.17 − 5.50i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.52 − 0.883i)3-s + (−0.249 + 0.433i)4-s + (0.411 + 0.713i)5-s − 1.24i·6-s + (0.0961 − 0.995i)7-s − 0.353·8-s + (1.06 + 1.83i)9-s + (−0.291 + 0.504i)10-s + (−1.16 − 0.673i)11-s + (0.764 − 0.441i)12-s − 0.826i·13-s + (0.643 − 0.293i)14-s − 1.45i·15-s + (−0.125 − 0.216i)16-s + (0.771 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543675 - 0.460296i\)
\(L(\frac12)\) \(\approx\) \(0.543675 - 0.460296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.254 + 2.63i)T \)
23 \( 1 + (-2.93 - 3.79i)T \)
good3 \( 1 + (2.64 + 1.52i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.920 - 1.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.87 + 2.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.97iT - 13T^{2} \)
17 \( 1 + (-3.17 + 5.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.80 + 6.59i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 3.95T + 29T^{2} \)
31 \( 1 + (-0.468 - 0.270i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.545 + 0.314i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.40iT - 41T^{2} \)
43 \( 1 - 5.38iT - 43T^{2} \)
47 \( 1 + (7.64 - 4.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.00463 - 0.00267i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.1 - 7.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.31 + 9.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.39 - 0.807i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.65T + 71T^{2} \)
73 \( 1 + (-7.69 - 4.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.47 + 2.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (-2.12 - 3.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−11.10631788745329358738363662882, −11.00174768772788853965139490598, −9.849538278904738672781605053299, −7.979694934622419342465697567893, −7.23420672761316478643116299416, −6.59653940321579453518907243168, −5.54249019791317943502059381766, −4.88718105859005485690619794204, −2.89398810949506200080876307207, −0.54614618050650743619397588456, 1.83683731662926946361019555194, 3.91801192079676648951341986111, 5.00324311159953675001395846621, 5.52859899381629045725518106765, 6.39359092034905562504140179494, 8.342232384835233584764286132628, 9.462312307127942395310577886098, 10.22572951257965332401241272465, 10.85370074393283827699360752207, 11.92737830373336536539487507622

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