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Properties

Label 2-1911-1911.1046-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.947 - 0.319i$
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  + (0.680 + 0.733i)3-s + (0.781 − 0.623i)4-s + (−0.997 + 0.0747i)7-s + (−0.0747 + 0.997i)9-s + (0.988 + 0.149i)12-s + (0.930 − 0.365i)13-s + (0.222 − 0.974i)16-s + (1.26 − 0.337i)19-s + (−0.733 − 0.680i)21-s + (0.997 + 0.0747i)25-s + (−0.781 + 0.623i)27-s + (−0.733 + 0.680i)28-s + (−1.91 + 0.514i)31-s + (0.563 + 0.826i)36-s + (0.794 − 0.0895i)37-s + ⋯
L(s)  = 1  + (0.680 + 0.733i)3-s + (0.781 − 0.623i)4-s + (−0.997 + 0.0747i)7-s + (−0.0747 + 0.997i)9-s + (0.988 + 0.149i)12-s + (0.930 − 0.365i)13-s + (0.222 − 0.974i)16-s + (1.26 − 0.337i)19-s + (−0.733 − 0.680i)21-s + (0.997 + 0.0747i)25-s + (−0.781 + 0.623i)27-s + (−0.733 + 0.680i)28-s + (−1.91 + 0.514i)31-s + (0.563 + 0.826i)36-s + (0.794 − 0.0895i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1046, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.640043474\)
\(L(\frac12)\) \(\approx\) \(1.640043474\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.680 - 0.733i)T \)
7 \( 1 + (0.997 - 0.0747i)T \)
13 \( 1 + (-0.930 + 0.365i)T \)
good2 \( 1 + (-0.781 + 0.623i)T^{2} \)
5 \( 1 + (-0.997 - 0.0747i)T^{2} \)
11 \( 1 + (0.149 + 0.988i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (-1.26 + 0.337i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.733 - 0.680i)T^{2} \)
31 \( 1 + (1.91 - 0.514i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.794 + 0.0895i)T + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (0.563 - 0.826i)T^{2} \)
43 \( 1 + (-0.587 - 1.90i)T + (-0.826 + 0.563i)T^{2} \)
47 \( 1 + (-0.149 - 0.988i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (0.433 - 0.900i)T^{2} \)
61 \( 1 + (1.84 - 0.722i)T + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (1.63 + 0.438i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.680 - 0.733i)T^{2} \)
73 \( 1 + (1.04 + 1.21i)T + (-0.149 + 0.988i)T^{2} \)
79 \( 1 + (0.781 + 1.35i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.781 - 0.623i)T^{2} \)
89 \( 1 + (0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.392 - 1.46i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.263774353406916764632347114837, −9.089689695395306063337914423013, −7.77297975565721670971076568045, −7.17597150950009081846085301167, −6.13244538937547941680994726367, −5.54406378107172663868863337199, −4.50652327069074776676688544683, −3.21058527854788805043951878611, −2.94246983278781845329826511525, −1.46158305537243108427254716072, 1.40059813044444084220214615126, 2.56692698158099585659625891670, 3.36183670548642569887958306736, 3.94698525373559206286430402667, 5.73113491756970435492734662523, 6.33532674271733005848791005007, 7.28411505492321268879650567153, 7.45384275486280907972432496074, 8.657481121929834831357036550604, 9.072779851267546006119591011689

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