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Properties

Label 2-1638-91.34-c1-0-6
Degree $2$
Conductor $1638$
Sign $-0.946 + 0.324i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.80 + 1.80i)5-s + (1.80 − 1.93i)7-s + (−0.707 + 0.707i)8-s − 2.54·10-s + (−0.936 + 0.936i)11-s + (−2 − 3i)13-s + (2.64 − 0.0951i)14-s − 1.00·16-s − 0.496·17-s + (−3.07 + 3.07i)19-s + (−1.80 − 1.80i)20-s − 1.32·22-s + 7.37i·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.806 + 0.806i)5-s + (0.681 − 0.732i)7-s + (−0.250 + 0.250i)8-s − 0.806·10-s + (−0.282 + 0.282i)11-s + (−0.554 − 0.832i)13-s + (0.706 − 0.0254i)14-s − 0.250·16-s − 0.120·17-s + (−0.704 + 0.704i)19-s + (−0.403 − 0.403i)20-s − 0.282·22-s + 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.946 + 0.324i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.946 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6611921940\)
\(L(\frac12)\) \(\approx\) \(0.6611921940\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-1.80 + 1.93i)T \)
13 \( 1 + (2 + 3i)T \)
good5 \( 1 + (1.80 - 1.80i)T - 5iT^{2} \)
11 \( 1 + (0.936 - 0.936i)T - 11iT^{2} \)
17 \( 1 + 0.496T + 17T^{2} \)
19 \( 1 + (3.07 - 3.07i)T - 19iT^{2} \)
23 \( 1 - 7.37iT - 23T^{2} \)
29 \( 1 + 7.25T + 29T^{2} \)
31 \( 1 + (5.73 - 5.73i)T - 31iT^{2} \)
37 \( 1 + (-3.54 + 3.54i)T - 37iT^{2} \)
41 \( 1 + (6.11 - 6.11i)T - 41iT^{2} \)
43 \( 1 + 6.85iT - 43T^{2} \)
47 \( 1 + (5.87 + 5.87i)T + 47iT^{2} \)
53 \( 1 - 4.26T + 53T^{2} \)
59 \( 1 + (1.37 + 1.37i)T + 59iT^{2} \)
61 \( 1 + 0.496iT - 61T^{2} \)
67 \( 1 + (-9.11 - 9.11i)T + 67iT^{2} \)
71 \( 1 + (11.3 + 11.3i)T + 71iT^{2} \)
73 \( 1 + (3.54 + 3.54i)T + 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (-7.87 + 7.87i)T - 83iT^{2} \)
89 \( 1 + (-10.2 - 10.2i)T + 89iT^{2} \)
97 \( 1 + (9.46 - 9.46i)T - 97iT^{2} \)
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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

−9.984345049100374435868782015613, −8.786867478005315380706150076848, −7.75257356675750716012866936320, −7.53611329896122314358015648426, −6.84113785687407694566533769248, −5.63909408521259725090482043990, −4.94593247711618898807991415631, −3.82610762567981973850775130247, −3.36164406387143574092386138345, −1.89910662698384420398297987110, 0.20030618940209709173334136044, 1.81880857804628856247717659919, 2.72147917061017320942033431986, 4.13816264990465228655178754093, 4.58652490982369269607991346499, 5.39376799773606935002493535129, 6.36813135332495611160721079097, 7.43007415235318825538323084040, 8.347679037790880052087515574208, 8.871886715446220414261387402964

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