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Properties

Label 2-1911-1911.1490-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.438 - 0.898i$
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  + (0.974 − 0.222i)3-s + (−0.781 + 0.623i)4-s + (−0.433 + 0.900i)7-s + (0.900 − 0.433i)9-s + (−0.623 + 0.781i)12-s + (0.781 + 0.623i)13-s + (0.222 − 0.974i)16-s + (−1.33 + 1.33i)19-s + (−0.222 + 0.974i)21-s + (0.433 + 0.900i)25-s + (0.781 − 0.623i)27-s + (−0.222 − 0.974i)28-s + (−0.158 + 0.158i)31-s + (−0.433 + 0.900i)36-s + (−0.0250 − 0.222i)37-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)3-s + (−0.781 + 0.623i)4-s + (−0.433 + 0.900i)7-s + (0.900 − 0.433i)9-s + (−0.623 + 0.781i)12-s + (0.781 + 0.623i)13-s + (0.222 − 0.974i)16-s + (−1.33 + 1.33i)19-s + (−0.222 + 0.974i)21-s + (0.433 + 0.900i)25-s + (0.781 − 0.623i)27-s + (−0.222 − 0.974i)28-s + (−0.158 + 0.158i)31-s + (−0.433 + 0.900i)36-s + (−0.0250 − 0.222i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1490, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270221887\)
\(L(\frac12)\) \(\approx\) \(1.270221887\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.974 + 0.222i)T \)
7 \( 1 + (0.433 - 0.900i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
good2 \( 1 + (0.781 - 0.623i)T^{2} \)
5 \( 1 + (-0.433 - 0.900i)T^{2} \)
11 \( 1 + (-0.781 + 0.623i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (1.33 - 1.33i)T - iT^{2} \)
23 \( 1 + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.158 - 0.158i)T - iT^{2} \)
37 \( 1 + (0.0250 + 0.222i)T + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (-0.433 - 0.900i)T^{2} \)
43 \( 1 + (-0.846 - 0.193i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.781 - 0.623i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.433 + 0.900i)T^{2} \)
61 \( 1 + (-0.974 - 0.777i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.752 + 0.752i)T + iT^{2} \)
71 \( 1 + (0.974 + 0.222i)T^{2} \)
73 \( 1 + (-1.87 - 0.656i)T + (0.781 + 0.623i)T^{2} \)
79 \( 1 + 1.56T + T^{2} \)
83 \( 1 + (0.781 + 0.623i)T^{2} \)
89 \( 1 + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (-0.467 + 0.467i)T - iT^{2} \)
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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

−9.301916514006840999169658147071, −8.636474873806619055215950092327, −8.318771292872946206160100124104, −7.35136726361606144034894704524, −6.46165058396030806966256924679, −5.55589617077667448847316641835, −4.30514245235482638159768512729, −3.70080273825724317257030065097, −2.82429853539119021995057176239, −1.71492037653187176914759643357, 0.908299208458257379552396556933, 2.37982215819696383206347969423, 3.54269066403874773466287038836, 4.25114651241328522102108424289, 4.93881211838393334127805443007, 6.17790058905080135797347723483, 6.89113594144458139684724203023, 7.924149956720301366988697171875, 8.623657852542217703581573523138, 9.168986907302255952870373961582

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