Properties

Label 2-1911-273.116-c0-0-2
Degree $2$
Conductor $1911$
Sign $-0.749 - 0.661i$
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  + (0.382 + 0.662i)2-s + (−0.5 + 0.866i)3-s + (0.207 − 0.358i)4-s + (0.923 + 1.60i)5-s − 0.765·6-s + 1.08·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (−0.382 + 0.662i)11-s + (0.207 + 0.358i)12-s − 13-s − 1.84·15-s + (0.207 + 0.358i)16-s + (0.382 − 0.662i)18-s + 0.765·20-s + ⋯
L(s)  = 1  + (0.382 + 0.662i)2-s + (−0.5 + 0.866i)3-s + (0.207 − 0.358i)4-s + (0.923 + 1.60i)5-s − 0.765·6-s + 1.08·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (−0.382 + 0.662i)11-s + (0.207 + 0.358i)12-s − 13-s − 1.84·15-s + (0.207 + 0.358i)16-s + (0.382 − 0.662i)18-s + 0.765·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.749 - 0.661i$
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: \((2,\ 1911,\ (\ :0),\ -0.749 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449917808\)
\(L(\frac12)\) \(\approx\) \(1.449917808\)
\(L(1)\) 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 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.84T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.84T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 0.765T + T^{2} \)
89 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.950756835600683855745170909590, −9.294800571696405405896255204477, −7.75200933715683022700033641635, −7.01092822703179509628583612825, −6.48696293143417147297671861677, −5.69622956218373760390498443685, −5.13256260481903109531878663629, −4.16973415007004942936615968804, −2.92375765469683277407709145621, −2.05611580767749605967054170809, 1.02268019764110511270924881952, 2.01993222046230178137897736561, 2.80085278337448965096891717605, 4.33889744536100124009003410657, 5.04949052705041863000786148130, 5.70844900249492314077503060587, 6.60707245181531177701951266639, 7.73643768018726526098922996859, 8.142932349533341060397994479852, 9.123578083768688105179514188344

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