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Properties

Label 2-1911-1911.1319-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.798 - 0.601i$
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  + (−0.294 − 0.955i)3-s + (−0.781 − 0.623i)4-s + (−0.563 + 0.826i)7-s + (−0.826 + 0.563i)9-s + (−0.365 + 0.930i)12-s + (0.149 + 0.988i)13-s + (0.222 + 0.974i)16-s + (−0.359 + 0.0962i)19-s + (0.955 + 0.294i)21-s + (0.563 + 0.826i)25-s + (0.781 + 0.623i)27-s + (0.955 − 0.294i)28-s + (0.216 − 0.0579i)31-s + (0.997 + 0.0747i)36-s + (−0.180 + 1.59i)37-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)3-s + (−0.781 − 0.623i)4-s + (−0.563 + 0.826i)7-s + (−0.826 + 0.563i)9-s + (−0.365 + 0.930i)12-s + (0.149 + 0.988i)13-s + (0.222 + 0.974i)16-s + (−0.359 + 0.0962i)19-s + (0.955 + 0.294i)21-s + (0.563 + 0.826i)25-s + (0.781 + 0.623i)27-s + (0.955 − 0.294i)28-s + (0.216 − 0.0579i)31-s + (0.997 + 0.0747i)36-s + (−0.180 + 1.59i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.798 - 0.601i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.798 - 0.601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5729282919\)
\(L(\frac12)\) \(\approx\) \(0.5729282919\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.294 + 0.955i)T \)
7 \( 1 + (0.563 - 0.826i)T \)
13 \( 1 + (-0.149 - 0.988i)T \)
good2 \( 1 + (0.781 + 0.623i)T^{2} \)
5 \( 1 + (-0.563 - 0.826i)T^{2} \)
11 \( 1 + (0.930 - 0.365i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 + (0.359 - 0.0962i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.955 + 0.294i)T^{2} \)
31 \( 1 + (-0.216 + 0.0579i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (0.180 - 1.59i)T + (-0.974 - 0.222i)T^{2} \)
41 \( 1 + (0.997 - 0.0747i)T^{2} \)
43 \( 1 + (0.766 + 0.825i)T + (-0.0747 + 0.997i)T^{2} \)
47 \( 1 + (-0.930 + 0.365i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.433 - 0.900i)T^{2} \)
61 \( 1 + (-0.108 - 0.722i)T + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (-1.02 - 0.275i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.294 + 0.955i)T^{2} \)
73 \( 1 + (-0.148 + 0.785i)T + (-0.930 - 0.365i)T^{2} \)
79 \( 1 + (-0.781 - 1.35i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.781 - 0.623i)T^{2} \)
89 \( 1 + (-0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.508 - 1.89i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.312639692466185579015193137215, −8.740494982403275617354217978546, −8.071359858284952980401046398532, −6.80244245889740282985614772723, −6.41981258610665641605337966466, −5.49875574836850286859451701921, −4.88087509138556039329842927567, −3.62811277323111581344692890630, −2.37677576170576559453695797951, −1.32035461672153633639479012433, 0.48325125750361904956094797860, 2.86798680180553918543620843063, 3.60531484064425320079602021306, 4.32190628703952717909646413924, 5.07710586233472010325865038023, 6.01025836840116232633862617318, 6.96046868843349172627696754593, 7.961637750419478476599795696141, 8.589752248958913704895208482733, 9.426191705234608290780535333744

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