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Properties

Label 2-1911-1911.1172-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.207 + 0.978i$
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 + (−0.149 + 0.988i)4-s + (0.0747 − 0.997i)7-s + (0.900 + 0.433i)9-s + (0.365 − 0.930i)12-s + (−0.680 − 0.733i)13-s + (−0.955 − 0.294i)16-s + (−1.38 − 1.38i)19-s + (−0.294 + 0.955i)21-s + (0.563 + 0.826i)25-s + (−0.781 − 0.623i)27-s + (0.974 + 0.222i)28-s + (1.91 − 0.514i)31-s + (−0.563 + 0.826i)36-s + (−0.955 − 0.705i)37-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)3-s + (−0.149 + 0.988i)4-s + (0.0747 − 0.997i)7-s + (0.900 + 0.433i)9-s + (0.365 − 0.930i)12-s + (−0.680 − 0.733i)13-s + (−0.955 − 0.294i)16-s + (−1.38 − 1.38i)19-s + (−0.294 + 0.955i)21-s + (0.563 + 0.826i)25-s + (−0.781 − 0.623i)27-s + (0.974 + 0.222i)28-s + (1.91 − 0.514i)31-s + (−0.563 + 0.826i)36-s + (−0.955 − 0.705i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.207 + 0.978i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.207 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5860855851\)
\(L(\frac12)\) \(\approx\) \(0.5860855851\)
\(L(1)\) 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 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (0.680 + 0.733i)T \)
good2 \( 1 + (0.149 - 0.988i)T^{2} \)
5 \( 1 + (-0.563 - 0.826i)T^{2} \)
11 \( 1 + (-0.781 - 0.623i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
23 \( 1 + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.733 + 0.680i)T^{2} \)
31 \( 1 + (-1.91 + 0.514i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.955 + 0.705i)T + (0.294 + 0.955i)T^{2} \)
41 \( 1 + (-0.563 - 0.826i)T^{2} \)
43 \( 1 + (-0.332 + 1.07i)T + (-0.826 - 0.563i)T^{2} \)
47 \( 1 + (-0.930 + 0.365i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (0.997 + 0.0747i)T^{2} \)
61 \( 1 + (0.571 - 0.455i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (-1.19 + 1.19i)T - iT^{2} \)
71 \( 1 + (-0.680 - 0.733i)T^{2} \)
73 \( 1 + (-0.340 + 1.80i)T + (-0.930 - 0.365i)T^{2} \)
79 \( 1 + (0.781 + 1.35i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.781 - 0.623i)T^{2} \)
89 \( 1 + (0.930 + 0.365i)T^{2} \)
97 \( 1 + (0.359 - 0.0962i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.191630716973750783740211397071, −8.294082244083058865268545932229, −7.45080083127106614027418213115, −7.02123402675307582454479399736, −6.21558982656584925588620106987, −4.89822610718736417887727571052, −4.52715996342155000460970807675, −3.44985234446650239457865128590, −2.26180102114178395023105969743, −0.50530244109433986070441117752, 1.41982465346476302525759630632, 2.48964129948780731861712496049, 4.18677861691045037514019631798, 4.81358699800191827772713837647, 5.56884093276622562542889597945, 6.39468960879905200609737161501, 6.69614753986448725328327108961, 8.190360444087781682451482789172, 8.869842491220620219621289450610, 9.955570678083141414915970169840

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