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Properties

Label 2-385-385.174-c0-0-0
Degree $2$
Conductor $385$
Sign $0.624 + 0.781i$
Analytic cond. $0.192140$
Root an. cond. $0.438337$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.500 − 1.53i)9-s + (0.309 − 0.951i)11-s + 1.61·12-s + (0.5 − 1.53i)13-s + (1.30 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.190 − 0.587i)17-s + (−0.309 + 0.951i)20-s − 1.61·21-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (−0.309 − 0.951i)28-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.500 − 1.53i)9-s + (0.309 − 0.951i)11-s + 1.61·12-s + (0.5 − 1.53i)13-s + (1.30 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.190 − 0.587i)17-s + (−0.309 + 0.951i)20-s − 1.61·21-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (−0.309 − 0.951i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(0.192140\)
Root analytic conductor: \(0.438337\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4509567790\)
\(L(\frac12)\) \(\approx\) \(0.4509567790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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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

−11.26382491551898448891999700078, −10.67862491160956418065115454251, −9.645394154221383949330667790611, −8.824976574840799768125466730449, −8.049676185820594677595570107483, −5.98240371678812572020781933343, −5.45006659013903298191116709414, −4.83590363791735742266352714364, −3.79877203436675633757995701105, −0.833600375793384608502220985143, 1.75974268084414129754728049890, 3.94234973679548375141285300852, 4.74270675637619621065310962214, 6.16004604623335553310086552764, 7.10685445628319700647681583510, 7.54621918637570984910776383611, 8.790348707883578236208205800983, 10.11636986004920285924557825587, 11.12679637278144364658854297242, 11.64800867766690871950379722489

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