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Properties

Label 2-322-161.45-c1-0-2
Degree $2$
Conductor $322$
Sign $0.691 - 0.722i$
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 + (1.05 + 0.609i)3-s + (−0.499 + 0.866i)4-s + (1.23 + 2.14i)5-s − 1.21i·6-s + (−1.54 + 2.14i)7-s + 0.999·8-s + (−0.756 − 1.31i)9-s + (1.23 − 2.14i)10-s + (1.09 + 0.633i)11-s + (−1.05 + 0.609i)12-s + 3.39i·13-s + (2.63 + 0.260i)14-s + 3.01i·15-s + (−0.5 − 0.866i)16-s + (−2.14 + 3.71i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.609 + 0.351i)3-s + (−0.249 + 0.433i)4-s + (0.552 + 0.957i)5-s − 0.497i·6-s + (−0.582 + 0.812i)7-s + 0.353·8-s + (−0.252 − 0.436i)9-s + (0.390 − 0.677i)10-s + (0.331 + 0.191i)11-s + (−0.304 + 0.175i)12-s + 0.941i·13-s + (0.703 + 0.0696i)14-s + 0.778i·15-s + (−0.125 − 0.216i)16-s + (−0.520 + 0.901i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.691 - 0.722i$
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: \((2,\ 322,\ (\ :1/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17898 + 0.503850i\)
\(L(\frac12)\) \(\approx\) \(1.17898 + 0.503850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 - 2.14i)T \)
23 \( 1 + (1.21 + 4.63i)T \)
good3 \( 1 + (-1.05 - 0.609i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.23 - 2.14i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.09 - 0.633i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.39iT - 13T^{2} \)
17 \( 1 + (2.14 - 3.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 - 2.14i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 8.99T + 29T^{2} \)
31 \( 1 + (-3.32 - 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.24 + 0.716i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.39iT - 41T^{2} \)
43 \( 1 + 12.9iT - 43T^{2} \)
47 \( 1 + (1.43 - 0.831i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.14 - 4.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.43 - 4.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.61 + 8.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (-3.32 - 1.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.44 + 3.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 + (3.43 + 5.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.49T + 97T^{2} \)
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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

−11.82965949618804559998049454734, −10.50325978203168522740991707695, −9.966620033231633139819555435971, −9.000581000537935887635227158877, −8.455641673305830250072210582507, −6.76127112562072248662552261566, −6.12498920094094489180102643742, −4.24924476222208548109163531543, −3.08093743873416203250719509907, −2.20213415081372846328326812245, 1.03420836307792054662078013489, 2.89878969317962650576651263275, 4.61994359330383168700813511172, 5.63713621662982688640151686431, 6.83994604037194320926037480113, 7.77843053805309997620544049155, 8.616284626801747682210067108185, 9.449202808534926968051915797754, 10.23056307811798848046383184867, 11.44583790365780658433625385455

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