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Properties

Label 2-1911-273.41-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.351 - 0.936i$
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 − 0.999·12-s + (−0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + (1.86 − 0.5i)19-s + i·25-s + 0.999i·27-s + (1 + i)31-s + (0.866 − 0.499i)36-s + (−0.133 + 0.5i)37-s + (0.499 − 0.866i)39-s + (−1.5 − 0.866i)43-s + (−0.866 − 0.499i)48-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 + 0.5i)4-s + (0.499 − 0.866i)9-s − 0.999·12-s + (−0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + (1.86 − 0.5i)19-s + i·25-s + 0.999i·27-s + (1 + i)31-s + (0.866 − 0.499i)36-s + (−0.133 + 0.5i)37-s + (0.499 − 0.866i)39-s + (−1.5 − 0.866i)43-s + (−0.866 − 0.499i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.351 - 0.936i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.351 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062757179\)
\(L(\frac12)\) \(\approx\) \(1.062757179\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.866 - 0.5i)T^{2} \)
5 \( 1 - iT^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.717100416923748641925735259226, −8.888570684920833738554163181261, −7.76568838220961176883609274819, −7.02413914767810770931155411016, −6.56103108927982318519861099276, −5.41422739010493702399411628258, −4.86677171930503993117300861643, −3.65754650675679707869145921483, −2.88770411271398253401517847738, −1.44964973556491401849265269260, 0.948001983444800233675880187368, 2.12312188782496237994429783950, 3.11147766457845559119372537524, 4.65088173344534236515405310674, 5.40750093563727167151294027270, 6.05834705408996204467785165507, 6.82583656161811514658994475278, 7.57288950542085024283797012652, 8.082306582284555386405282219231, 9.662649194086094860800922068517

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