Properties

Label 2-1911-1911.1601-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.997 - 0.0703i$
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  + (−0.563 − 0.826i)3-s + (−0.974 + 0.222i)4-s + (−0.930 + 0.365i)7-s + (−0.365 + 0.930i)9-s + (0.733 + 0.680i)12-s + (−0.294 − 0.955i)13-s + (0.900 − 0.433i)16-s + (−0.206 + 0.772i)19-s + (0.826 + 0.563i)21-s + (0.930 + 0.365i)25-s + (0.974 − 0.222i)27-s + (0.826 − 0.563i)28-s + (−0.438 + 1.63i)31-s + (0.149 − 0.988i)36-s + (1.49 − 0.940i)37-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)3-s + (−0.974 + 0.222i)4-s + (−0.930 + 0.365i)7-s + (−0.365 + 0.930i)9-s + (0.733 + 0.680i)12-s + (−0.294 − 0.955i)13-s + (0.900 − 0.433i)16-s + (−0.206 + 0.772i)19-s + (0.826 + 0.563i)21-s + (0.930 + 0.365i)25-s + (0.974 − 0.222i)27-s + (0.826 − 0.563i)28-s + (−0.438 + 1.63i)31-s + (0.149 − 0.988i)36-s + (1.49 − 0.940i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.997 - 0.0703i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.997 - 0.0703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5775468446\)
\(L(\frac12)\) \(\approx\) \(0.5775468446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.563 + 0.826i)T \)
7 \( 1 + (0.930 - 0.365i)T \)
13 \( 1 + (0.294 + 0.955i)T \)
good2 \( 1 + (0.974 - 0.222i)T^{2} \)
5 \( 1 + (-0.930 - 0.365i)T^{2} \)
11 \( 1 + (0.680 + 0.733i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 + (0.206 - 0.772i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (-0.826 + 0.563i)T^{2} \)
31 \( 1 + (0.438 - 1.63i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (-1.49 + 0.940i)T + (0.433 - 0.900i)T^{2} \)
41 \( 1 + (0.149 + 0.988i)T^{2} \)
43 \( 1 + (-1.85 - 0.139i)T + (0.988 + 0.149i)T^{2} \)
47 \( 1 + (-0.680 - 0.733i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (0.781 + 0.623i)T^{2} \)
61 \( 1 + (-0.432 - 1.40i)T + (-0.826 + 0.563i)T^{2} \)
67 \( 1 + (-0.488 - 1.82i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.563 + 0.826i)T^{2} \)
73 \( 1 + (0.799 + 1.83i)T + (-0.680 + 0.733i)T^{2} \)
79 \( 1 + (-0.974 + 1.68i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.974 + 0.222i)T^{2} \)
89 \( 1 + (-0.974 - 0.222i)T^{2} \)
97 \( 1 + (-1.77 + 0.474i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.243193346398021406613723627543, −8.667134484569083415372921101404, −7.75262841182234544421579492385, −7.16020614947644678192126739408, −6.04156938293624426987808665814, −5.57324466198587284941521911202, −4.65811131764257318747413394117, −3.48031050487775218882478980031, −2.57459050455558449000182842289, −0.916018502266066632276688399408, 0.66200065159909362692704804930, 2.72699782388362268679401445283, 3.94166112762753124665988402406, 4.36317586304844819470952779904, 5.23565027978708710594763965252, 6.16778747928853324681873921742, 6.76459255122640045900276777931, 7.939808154617566105532523218142, 9.041668459817913007639074913878, 9.432516117578416153890256800006

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