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Properties

Label 2-1911-1911.1793-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.284 + 0.958i$
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  + (0.222 + 0.974i)3-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)7-s + (−0.900 + 0.433i)9-s + (0.623 − 0.781i)12-s + (0.222 − 0.974i)13-s + (−0.222 + 0.974i)16-s − 0.867i·19-s + (0.222 − 0.974i)21-s + (0.900 − 0.433i)25-s + (−0.623 − 0.781i)27-s + (0.222 + 0.974i)28-s − 1.56i·31-s + (0.900 + 0.433i)36-s + (−1.22 − 0.974i)37-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)7-s + (−0.900 + 0.433i)9-s + (0.623 − 0.781i)12-s + (0.222 − 0.974i)13-s + (−0.222 + 0.974i)16-s − 0.867i·19-s + (0.222 − 0.974i)21-s + (0.900 − 0.433i)25-s + (−0.623 − 0.781i)27-s + (0.222 + 0.974i)28-s − 1.56i·31-s + (0.900 + 0.433i)36-s + (−1.22 − 0.974i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7078436431\)
\(L(\frac12)\) \(\approx\) \(0.7078436431\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
good2 \( 1 + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + 0.867iT - T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + 1.56iT - T^{2} \)
37 \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.900 - 0.433i)T^{2} \)
61 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + 1.94iT - T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.678 - 1.40i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + 1.24T + T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T^{2} \)
97 \( 1 - 0.867iT - T^{2} \)
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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

−9.215880669452303808397119005360, −8.850001229405004371983541913245, −7.82560581529999398345311329571, −6.73937679846215164851542708970, −5.81028471259674975943899546579, −5.20581480491753057806433560051, −4.25663410994923618895804826789, −3.55107193131529869487269428697, −2.48644845506856831680176616664, −0.52015482043541351460326387282, 1.54452901296351434954615398343, 2.92018213559089562099993325316, 3.43600009006879558460643817162, 4.62024973683610817110241332834, 5.69664216271924606355725246671, 6.64080123554589201993123860109, 7.10797671251881982200861291433, 8.113376120606596084347978925418, 8.736545866439784829212463584223, 9.235562248413805981504682517499

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