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Properties

Label 2-1911-39.35-c0-0-5
Degree $2$
Conductor $1911$
Sign $-0.711 + 0.702i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
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  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)19-s + 25-s − 0.999·27-s − 2·31-s + (−0.499 + 0.866i)36-s + (0.5 − 0.866i)37-s + (−0.499 − 0.866i)39-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)48-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)19-s + 25-s − 0.999·27-s − 2·31-s + (−0.499 + 0.866i)36-s + (0.5 − 0.866i)37-s + (−0.499 − 0.866i)39-s + (0.5 + 0.866i)43-s + (0.499 + 0.866i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (932, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075277547\)
\(L(\frac12)\) \(\approx\) \(1.075277547\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.055138029617111243271782938313, −8.433519947829370315942967901258, −7.54597049667449703388064285091, −6.72656206201205974021336890579, −5.94312688844674896336377359163, −5.23254816683026408419955155472, −4.11291657171887009703308756930, −3.04066019187052892660605350481, −1.93051786631205222761730056941, −0.76093550577132874140630587934, 2.02481444343192156444126202768, 3.21554307014154957171017323561, 3.87767303691356768798776058003, 4.57355223528254881244179533414, 5.46855858750128013691048401194, 6.61266398511722379631599483924, 7.61256818486104446635497022027, 8.251738121883670363180207639903, 9.104006363988677245194226858925, 9.281178927848861083330441581942

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