Properties

Label 2-1911-273.116-c0-0-3
Degree $2$
Conductor $1911$
Sign $0.0725 + 0.997i$
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  + (−0.923 − 1.60i)2-s + (−0.5 + 0.866i)3-s + (−1.20 + 2.09i)4-s + (0.382 + 0.662i)5-s + 1.84·6-s + 2.61·8-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (0.923 − 1.60i)11-s + (−1.20 − 2.09i)12-s − 13-s − 0.765·15-s + (−1.20 − 2.09i)16-s + (−0.923 + 1.60i)18-s − 1.84·20-s + ⋯
L(s)  = 1  + (−0.923 − 1.60i)2-s + (−0.5 + 0.866i)3-s + (−1.20 + 2.09i)4-s + (0.382 + 0.662i)5-s + 1.84·6-s + 2.61·8-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)10-s + (0.923 − 1.60i)11-s + (−1.20 − 2.09i)12-s − 13-s − 0.765·15-s + (−1.20 − 2.09i)16-s + (−0.923 + 1.60i)18-s − 1.84·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.0725 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5524889482\)
\(L(\frac12)\) \(\approx\) \(0.5524889482\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 0.765T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.84T + T^{2} \)
89 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.565739629424013785584287272319, −8.802730297853633397307169627209, −8.154656968030285805445040387504, −6.87566564577993467265032906115, −5.98977845178304418029537669471, −4.85755724694160315751770063013, −3.79875805411527859739085903414, −3.22116140259266537153710783466, −2.29521423180542013200533193948, −0.73179004147242198548520352878, 1.11076249765841372462635910664, 2.09518598820693393259355413120, 4.50176805905612089524876100345, 5.06134951675157196760628294721, 5.86143835199875297170447651456, 6.68408803780852778874254812385, 7.22241996469806786467004795754, 7.75353019627381015154804703171, 8.680848409828330737293405755658, 9.415207071886573406393934987956

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