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Properties

Label 2-385-385.69-c0-0-0
Degree $2$
Conductor $385$
Sign $0.794 + 0.606i$
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  + (−0.190 − 0.587i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.5 − 0.363i)9-s + (−0.809 + 0.587i)11-s − 0.618·12-s + (0.5 − 0.363i)13-s + (0.190 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−1.30 − 0.951i)17-s + (0.809 − 0.587i)20-s + 0.618·21-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.809 + 0.587i)28-s + ⋯
L(s)  = 1  + (−0.190 − 0.587i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.309 + 0.951i)7-s + (0.5 − 0.363i)9-s + (−0.809 + 0.587i)11-s − 0.618·12-s + (0.5 − 0.363i)13-s + (0.190 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−1.30 − 0.951i)17-s + (0.809 − 0.587i)20-s + 0.618·21-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (0.809 + 0.587i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(0.192140\)
Root analytic conductor: \(0.438337\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 385,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8928358410\)
\(L(\frac12)\) \(\approx\) \(0.8928358410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.29773144299456552844964391632, −10.59452695803254149514756952258, −9.645987168810211145052907234681, −9.027236053733943027857408693865, −7.34588172097062492886993454137, −6.61329261852474504757185702650, −5.86980915098489462237671932939, −4.94626172009704051701632113508, −2.79125159704058621730022759357, −1.79356564242978974157212653667, 2.14514729022346113954780542835, 3.80196035183972551409375030408, 4.53700512658918026986089620041, 5.91200288383092176263042494801, 6.93603667954870709474953024404, 8.040689172827787187665712309246, 8.865832532871213229761023422465, 9.981566790290453144002490608533, 10.69563414178761088713617864378, 11.44281190140923863459442884696

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