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Properties

Label 2-1911-1911.1181-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.893 - 0.449i$
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  + (0.997 + 0.0747i)3-s + (−0.974 − 0.222i)4-s + (0.149 + 0.988i)7-s + (0.988 + 0.149i)9-s + (−0.955 − 0.294i)12-s + (−0.680 − 0.733i)13-s + (0.900 + 0.433i)16-s + (1.14 + 0.307i)19-s + (0.0747 + 0.997i)21-s + (−0.149 + 0.988i)25-s + (0.974 + 0.222i)27-s + (0.0747 − 0.997i)28-s + (1.63 + 0.438i)31-s + (−0.930 − 0.365i)36-s + (−0.0633 − 0.0397i)37-s + ⋯
L(s)  = 1  + (0.997 + 0.0747i)3-s + (−0.974 − 0.222i)4-s + (0.149 + 0.988i)7-s + (0.988 + 0.149i)9-s + (−0.955 − 0.294i)12-s + (−0.680 − 0.733i)13-s + (0.900 + 0.433i)16-s + (1.14 + 0.307i)19-s + (0.0747 + 0.997i)21-s + (−0.149 + 0.988i)25-s + (0.974 + 0.222i)27-s + (0.0747 − 0.997i)28-s + (1.63 + 0.438i)31-s + (−0.930 − 0.365i)36-s + (−0.0633 − 0.0397i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.893 - 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.356585055\)
\(L(\frac12)\) \(\approx\) \(1.356585055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.997 - 0.0747i)T \)
7 \( 1 + (-0.149 - 0.988i)T \)
13 \( 1 + (0.680 + 0.733i)T \)
good2 \( 1 + (0.974 + 0.222i)T^{2} \)
5 \( 1 + (0.149 - 0.988i)T^{2} \)
11 \( 1 + (0.294 + 0.955i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + (-1.14 - 0.307i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.0747 + 0.997i)T^{2} \)
31 \( 1 + (-1.63 - 0.438i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.0633 + 0.0397i)T + (0.433 + 0.900i)T^{2} \)
41 \( 1 + (-0.930 + 0.365i)T^{2} \)
43 \( 1 + (-0.167 - 0.246i)T + (-0.365 + 0.930i)T^{2} \)
47 \( 1 + (-0.294 - 0.955i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.781 - 0.623i)T^{2} \)
61 \( 1 + (1.29 + 1.40i)T + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (1.82 - 0.488i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.997 - 0.0747i)T^{2} \)
73 \( 1 + (-0.554 - 0.751i)T + (-0.294 + 0.955i)T^{2} \)
79 \( 1 + (-0.974 + 1.68i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.974 - 0.222i)T^{2} \)
89 \( 1 + (-0.974 + 0.222i)T^{2} \)
97 \( 1 + (-0.416 - 1.55i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.429061245477638757396733138025, −8.738951943807357182124324619475, −8.025908313236281981249577966608, −7.46538516183528783233178797176, −6.15016046551087951563781219680, −5.21323803513305838429610292510, −4.66650804333502533447255214780, −3.45649051231677615840108943257, −2.77441337761518908639167188373, −1.44009107809052135572790220226, 1.09978296947111078261497139065, 2.57890300705227435766041271927, 3.56415699572555229382312085957, 4.39959508306519636554317153117, 4.85854367225361979173462313586, 6.32561732074206196491561834247, 7.34458477419780686190882898766, 7.76246742893274922159139831856, 8.580882930918848088714915447735, 9.327297393944913985847715722759

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