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Properties

Label 2-1911-1911.1571-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.744 - 0.667i$
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  + (0.433 + 0.900i)3-s + (0.294 + 0.955i)4-s + (−0.988 + 0.149i)7-s + (−0.623 + 0.781i)9-s + (−0.733 + 0.680i)12-s + (0.997 − 0.0747i)13-s + (−0.826 + 0.563i)16-s + (−1.29 + 1.29i)19-s + (−0.563 − 0.826i)21-s + (0.930 − 0.365i)25-s + (−0.974 − 0.222i)27-s + (−0.433 − 0.900i)28-s + (−0.275 − 1.02i)31-s + (−0.930 − 0.365i)36-s + (−0.826 + 0.436i)37-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)3-s + (0.294 + 0.955i)4-s + (−0.988 + 0.149i)7-s + (−0.623 + 0.781i)9-s + (−0.733 + 0.680i)12-s + (0.997 − 0.0747i)13-s + (−0.826 + 0.563i)16-s + (−1.29 + 1.29i)19-s + (−0.563 − 0.826i)21-s + (0.930 − 0.365i)25-s + (−0.974 − 0.222i)27-s + (−0.433 − 0.900i)28-s + (−0.275 − 1.02i)31-s + (−0.930 − 0.365i)36-s + (−0.826 + 0.436i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146128915\)
\(L(\frac12)\) \(\approx\) \(1.146128915\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.433 - 0.900i)T \)
7 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (-0.997 + 0.0747i)T \)
good2 \( 1 + (-0.294 - 0.955i)T^{2} \)
5 \( 1 + (-0.930 + 0.365i)T^{2} \)
11 \( 1 + (-0.974 - 0.222i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (1.29 - 1.29i)T - iT^{2} \)
23 \( 1 + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (-0.0747 + 0.997i)T^{2} \)
31 \( 1 + (0.275 + 1.02i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.826 - 0.436i)T + (0.563 - 0.826i)T^{2} \)
41 \( 1 + (-0.930 + 0.365i)T^{2} \)
43 \( 1 + (-1.04 - 1.53i)T + (-0.365 + 0.930i)T^{2} \)
47 \( 1 + (-0.680 + 0.733i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (0.149 + 0.988i)T^{2} \)
61 \( 1 + (-1.42 + 0.326i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.467 - 0.467i)T + iT^{2} \)
71 \( 1 + (0.997 - 0.0747i)T^{2} \)
73 \( 1 + (-0.0299 + 0.0685i)T + (-0.680 - 0.733i)T^{2} \)
79 \( 1 + (0.974 + 1.68i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.974 - 0.222i)T^{2} \)
89 \( 1 + (0.680 + 0.733i)T^{2} \)
97 \( 1 + (0.206 + 0.772i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.659805618315734722913343214338, −8.724309660616252419214843108128, −8.417340127547276457093796827529, −7.50813098952926875352880198567, −6.43237967944058520792122820662, −5.86518660131462415618933441116, −4.48201991745781416441553944319, −3.77720906163101894934892520845, −3.14974065762054075026438762729, −2.18343571145808383793600593133, 0.77481589006488300668604388096, 2.01766061956006741958847214377, 2.94159529771980642571102212625, 3.98453846710987714957742149208, 5.31486662720656806971095180486, 6.11160331890744476739774440741, 6.89945012523044897096214119099, 7.06194887887006452531473220481, 8.654306348691335625845036043874, 8.857017846492122911311276907102

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