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Properties

Label 2-322-161.160-c1-0-15
Degree $2$
Conductor $322$
Sign $-0.728 + 0.684i$
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  + 2-s − 3.02i·3-s + 4-s − 2·5-s − 3.02i·6-s + (−2.56 − 0.662i)7-s + 8-s − 6.12·9-s − 2·10-s − 3.02i·11-s − 3.02i·12-s + 1.69i·13-s + (−2.56 − 0.662i)14-s + 6.04i·15-s + 16-s + 7.12·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.74i·3-s + 0.5·4-s − 0.894·5-s − 1.23i·6-s + (−0.968 − 0.250i)7-s + 0.353·8-s − 2.04·9-s − 0.632·10-s − 0.910i·11-s − 0.871i·12-s + 0.470i·13-s + (−0.684 − 0.176i)14-s + 1.55i·15-s + 0.250·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533470 - 1.34663i\)
\(L(\frac12)\) \(\approx\) \(0.533470 - 1.34663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + (2.56 + 0.662i)T \)
23 \( 1 + (-2.56 + 4.05i)T \)
good3 \( 1 + 3.02iT - 3T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 - 1.69iT - 13T^{2} \)
17 \( 1 - 7.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8.10iT - 31T^{2} \)
37 \( 1 + 1.69iT - 37T^{2} \)
41 \( 1 + 3.39iT - 41T^{2} \)
43 \( 1 + 3.02iT - 43T^{2} \)
47 \( 1 - 7.36iT - 47T^{2} \)
53 \( 1 - 13.7iT - 53T^{2} \)
59 \( 1 - 9.06iT - 59T^{2} \)
61 \( 1 + 4.24T + 61T^{2} \)
67 \( 1 - 0.371iT - 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 3.97iT - 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−11.82776699975037832139738663596, −10.76420145334003286719694835397, −9.221452561807388626461591641302, −7.82352221826085814684061084194, −7.47950501562766099494543993519, −6.38633885221180702233745407687, −5.65495942503521892204232608008, −3.75399559841382008852367090320, −2.78252869691569325060875633182, −0.863144402808129779229186426302, 3.28817301954301117711180868348, 3.56189248689979180419366785891, 4.92389216227506731130061848997, 5.57865861726097544518528573132, 7.11357815209502445875749596586, 8.246959339265266843378266238274, 9.732374449302687945881902930171, 9.877863475371695219259409470648, 11.09704286163543972026624220765, 11.93301164861121264666560866469

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