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Properties

Label 2-322-161.160-c1-0-10
Degree $2$
Conductor $322$
Sign $0.305 - 0.952i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.305 - 0.952i$
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: \((2,\ 322,\ (\ :1/2),\ 0.305 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82044 + 1.32771i\)
\(L(\frac12)\) \(\approx\) \(1.82044 + 1.32771i\)
\(L(\frac{3}{2})\) 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

−11.38619567134518594122798374488, −10.82489369135487177475214367912, −10.30126389398245167341632079909, −9.024709042469512414112679283315, −8.406414333879055621476288266889, −6.56507710054688070249345860811, −5.48320614247508844034154108789, −4.76301756899638923550747388137, −3.83642081542673910127861349121, −2.43512090514490396848984430702, 1.86656986953554832765310428626, 2.22456787595562881073116324856, 4.50037387115595650242597798419, 5.71140972642038346747114610085, 6.55815224023800587509397866822, 7.32572411231876420151329836142, 8.333948529591292030549002830157, 9.432168058379395952569899735935, 11.04476468105532138827434311565, 11.60598591620432577967675910458

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