Properties

Label 2-1911-273.164-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.986 - 0.165i$
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  + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)4-s + (0.499 + 0.866i)9-s + (−0.499 − 0.866i)12-s i·13-s + (0.499 + 0.866i)16-s + (1.36 + 0.366i)19-s + (0.866 + 0.5i)25-s + 0.999i·27-s + (−0.366 − 1.36i)31-s − 0.999i·36-s + (1.36 + 0.366i)37-s + (0.5 − 0.866i)39-s + 0.999i·48-s + (−0.5 + 0.866i)52-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)4-s + (0.499 + 0.866i)9-s + (−0.499 − 0.866i)12-s i·13-s + (0.499 + 0.866i)16-s + (1.36 + 0.366i)19-s + (0.866 + 0.5i)25-s + 0.999i·27-s + (−0.366 − 1.36i)31-s − 0.999i·36-s + (1.36 + 0.366i)37-s + (0.5 − 0.866i)39-s + 0.999i·48-s + (−0.5 + 0.866i)52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.334004760\)
\(L(\frac12)\) \(\approx\) \(1.334004760\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.477341338259458893306394282814, −8.760607957317663891197463693088, −7.956339498961771084502262398126, −7.40476030525170037811442556848, −5.97925185164848238615695661730, −5.27550245125876390201253947971, −4.48906863068299185180883496104, −3.59161207848206160350047328400, −2.74672940440215880259814618717, −1.24684666355824255855974736517, 1.22806627163669616304146127096, 2.64783293934053103488142933070, 3.44671643779255641204094163171, 4.33857934541090367550070505772, 5.15394308420142615881614614197, 6.43564896665599533813345617299, 7.22087102442766406190290067977, 7.86386597940860429570980764551, 8.667809404409515792891162518130, 9.254176189380290390276504021849

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