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Properties

Label 2-1911-1911.380-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.807 - 0.590i$
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  + (0.955 + 0.294i)3-s + (0.623 + 0.781i)4-s + (0.826 − 0.563i)7-s + (0.826 + 0.563i)9-s + (0.365 + 0.930i)12-s + (−0.988 − 0.149i)13-s + (−0.222 + 0.974i)16-s + (−0.826 − 1.43i)19-s + (0.955 − 0.294i)21-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)27-s + (0.955 + 0.294i)28-s + (−0.623 − 1.07i)31-s + (0.0747 + 0.997i)36-s + (−1.23 + 1.54i)37-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)3-s + (0.623 + 0.781i)4-s + (0.826 − 0.563i)7-s + (0.826 + 0.563i)9-s + (0.365 + 0.930i)12-s + (−0.988 − 0.149i)13-s + (−0.222 + 0.974i)16-s + (−0.826 − 1.43i)19-s + (0.955 − 0.294i)21-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)27-s + (0.955 + 0.294i)28-s + (−0.623 − 1.07i)31-s + (0.0747 + 0.997i)36-s + (−1.23 + 1.54i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.807 - 0.590i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.807 - 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.905054021\)
\(L(\frac12)\) \(\approx\) \(1.905054021\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.955 - 0.294i)T \)
7 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
good2 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.955 - 0.294i)T^{2} \)
31 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.23 - 1.54i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 + (-0.0747 + 0.997i)T^{2} \)
43 \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \)
47 \( 1 + (-0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.733 + 0.680i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.955 + 0.294i)T^{2} \)
73 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.317534797912828790676755453707, −8.526187671267073075425034359958, −8.012673966429348336861529913455, −7.11795563804252976738666561721, −6.84822311249560426396658941291, −5.13338929979716032932364470718, −4.47466581679816967531617889735, −3.53869103898367164607431209370, −2.64315324205193256685826672558, −1.81038035462265313135256670360, 1.59923147802437042794207330508, 2.15674457439805858233442287041, 3.19232645614706351986389483604, 4.46978758032468489260300218633, 5.29136811824406021589257658749, 6.24175867962169787581865165009, 7.05744869386896324346486685560, 7.75112463548024962321189478871, 8.567341408834525379089033194449, 9.196104738879872679370390033153

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