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Properties

Label 2-1911-1911.1151-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.692 + 0.721i$
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  + (0.149 − 0.988i)3-s + (0.433 − 0.900i)4-s + (−0.294 − 0.955i)7-s + (−0.955 − 0.294i)9-s + (−0.826 − 0.563i)12-s + (0.997 + 0.0747i)13-s + (−0.623 − 0.781i)16-s + (0.241 + 0.902i)19-s + (−0.988 + 0.149i)21-s + (0.294 − 0.955i)25-s + (−0.433 + 0.900i)27-s + (−0.988 − 0.149i)28-s + (−0.170 − 0.638i)31-s + (−0.680 + 0.733i)36-s + (−0.430 + 1.23i)37-s + ⋯
L(s)  = 1  + (0.149 − 0.988i)3-s + (0.433 − 0.900i)4-s + (−0.294 − 0.955i)7-s + (−0.955 − 0.294i)9-s + (−0.826 − 0.563i)12-s + (0.997 + 0.0747i)13-s + (−0.623 − 0.781i)16-s + (0.241 + 0.902i)19-s + (−0.988 + 0.149i)21-s + (0.294 − 0.955i)25-s + (−0.433 + 0.900i)27-s + (−0.988 − 0.149i)28-s + (−0.170 − 0.638i)31-s + (−0.680 + 0.733i)36-s + (−0.430 + 1.23i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.692 + 0.721i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.239433313\)
\(L(\frac12)\) \(\approx\) \(1.239433313\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.149 + 0.988i)T \)
7 \( 1 + (0.294 + 0.955i)T \)
13 \( 1 + (-0.997 - 0.0747i)T \)
good2 \( 1 + (-0.433 + 0.900i)T^{2} \)
5 \( 1 + (-0.294 + 0.955i)T^{2} \)
11 \( 1 + (0.563 + 0.826i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + (-0.241 - 0.902i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (0.988 + 0.149i)T^{2} \)
31 \( 1 + (0.170 + 0.638i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.430 - 1.23i)T + (-0.781 - 0.623i)T^{2} \)
41 \( 1 + (-0.680 - 0.733i)T^{2} \)
43 \( 1 + (0.548 + 0.215i)T + (0.733 + 0.680i)T^{2} \)
47 \( 1 + (-0.563 - 0.826i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (0.974 + 0.222i)T^{2} \)
61 \( 1 + (-1.64 - 0.123i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.514 - 1.91i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.149 + 0.988i)T^{2} \)
73 \( 1 + (0.173 + 0.328i)T + (-0.563 + 0.826i)T^{2} \)
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 \( 1 + (-1.70 - 0.457i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.032616121383667173984995513865, −8.209434646559580699614835776880, −7.43078055615819831054294581854, −6.61860578126551377081683720592, −6.20451817036785674050077435740, −5.31187039026756286850959854811, −4.05746148875060624481099831798, −3.02234073906810686931004854526, −1.81104594810244583942966847628, −0.916236871481454022994742601021, 2.13944282631257454673267406759, 3.15732450418247743070201678558, 3.63082097653915010206692574240, 4.79001983110800603333270110990, 5.62544787337852186777423818828, 6.48357375986232961348635143954, 7.40850375500321669614201480179, 8.431852997525680539446563199392, 8.852315336421182907545131202085, 9.462139577085635724683840921114

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