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Properties

Label 2-1911-1911.332-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.957 - 0.288i$
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  + (0.930 + 0.365i)3-s + (−0.433 − 0.900i)4-s + (0.680 + 0.733i)7-s + (0.733 + 0.680i)9-s + (−0.0747 − 0.997i)12-s + (0.563 + 0.826i)13-s + (−0.623 + 0.781i)16-s + (−0.0193 − 0.0722i)19-s + (0.365 + 0.930i)21-s + (−0.680 + 0.733i)25-s + (0.433 + 0.900i)27-s + (0.365 − 0.930i)28-s + (−0.488 − 1.82i)31-s + (0.294 − 0.955i)36-s + (0.350 − 0.122i)37-s + ⋯
L(s)  = 1  + (0.930 + 0.365i)3-s + (−0.433 − 0.900i)4-s + (0.680 + 0.733i)7-s + (0.733 + 0.680i)9-s + (−0.0747 − 0.997i)12-s + (0.563 + 0.826i)13-s + (−0.623 + 0.781i)16-s + (−0.0193 − 0.0722i)19-s + (0.365 + 0.930i)21-s + (−0.680 + 0.733i)25-s + (0.433 + 0.900i)27-s + (0.365 − 0.930i)28-s + (−0.488 − 1.82i)31-s + (0.294 − 0.955i)36-s + (0.350 − 0.122i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.957 - 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.566507062\)
\(L(\frac12)\) \(\approx\) \(1.566507062\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.930 - 0.365i)T \)
7 \( 1 + (-0.680 - 0.733i)T \)
13 \( 1 + (-0.563 - 0.826i)T \)
good2 \( 1 + (0.433 + 0.900i)T^{2} \)
5 \( 1 + (0.680 - 0.733i)T^{2} \)
11 \( 1 + (0.997 + 0.0747i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + (0.0193 + 0.0722i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.365 + 0.930i)T^{2} \)
31 \( 1 + (0.488 + 1.82i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.350 + 0.122i)T + (0.781 - 0.623i)T^{2} \)
41 \( 1 + (0.294 + 0.955i)T^{2} \)
43 \( 1 + (-0.202 + 1.34i)T + (-0.955 - 0.294i)T^{2} \)
47 \( 1 + (-0.997 - 0.0747i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (-0.974 + 0.222i)T^{2} \)
61 \( 1 + (-0.0841 - 0.123i)T + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.0579 + 0.216i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.930 - 0.365i)T^{2} \)
73 \( 1 + (0.0487 + 1.30i)T + (-0.997 + 0.0747i)T^{2} \)
79 \( 1 + (-0.433 - 0.751i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.433 - 0.900i)T^{2} \)
89 \( 1 + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.344658473687988635551910964713, −8.829976978397936954542262848587, −8.103596316283676749813707558651, −7.21625309633916111177597773098, −6.06359372606088859661038136923, −5.36775889926045371019122338499, −4.45252517450846087671208819158, −3.77867679970053553096661581190, −2.33589171466143036792020811862, −1.61638763587599945610963593872, 1.26345127330152227728515305039, 2.62540963773388752522455478226, 3.52217190462514761833408791018, 4.17745413571124617293385865406, 5.11163089870509055157683312980, 6.45685416181522348569660883662, 7.29875481891438310350947615214, 7.981490977904325492413587005171, 8.344334784540933312518323290251, 9.129459038459328365453751528234

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