Properties

Label 2-1911-1911.1490-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.00548 - 0.999i$
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  + (−0.974 + 0.222i)3-s + (−0.781 + 0.623i)4-s + (−0.900 − 0.433i)7-s + (0.900 − 0.433i)9-s + (0.623 − 0.781i)12-s + (0.974 + 0.222i)13-s + (0.222 − 0.974i)16-s + (0.467 − 0.467i)19-s + (0.974 + 0.222i)21-s + (0.433 + 0.900i)25-s + (−0.781 + 0.623i)27-s + (0.974 − 0.222i)28-s + (−1.40 + 1.40i)31-s + (−0.433 + 0.900i)36-s + (0.222 + 1.97i)37-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)3-s + (−0.781 + 0.623i)4-s + (−0.900 − 0.433i)7-s + (0.900 − 0.433i)9-s + (0.623 − 0.781i)12-s + (0.974 + 0.222i)13-s + (0.222 − 0.974i)16-s + (0.467 − 0.467i)19-s + (0.974 + 0.222i)21-s + (0.433 + 0.900i)25-s + (−0.781 + 0.623i)27-s + (0.974 − 0.222i)28-s + (−1.40 + 1.40i)31-s + (−0.433 + 0.900i)36-s + (0.222 + 1.97i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00548 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00548 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.00548 - 0.999i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1490, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.00548 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5231231073\)
\(L(\frac12)\) \(\approx\) \(0.5231231073\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.974 - 0.222i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.974 - 0.222i)T \)
good2 \( 1 + (0.781 - 0.623i)T^{2} \)
5 \( 1 + (-0.433 - 0.900i)T^{2} \)
11 \( 1 + (-0.781 + 0.623i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (-0.467 + 0.467i)T - iT^{2} \)
23 \( 1 + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
37 \( 1 + (-0.222 - 1.97i)T + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (-0.433 - 0.900i)T^{2} \)
43 \( 1 + (0.846 + 0.193i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.781 - 0.623i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.433 + 0.900i)T^{2} \)
61 \( 1 + (0.974 + 0.777i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (-1.19 - 1.19i)T + iT^{2} \)
71 \( 1 + (0.974 + 0.222i)T^{2} \)
73 \( 1 + (-0.211 - 0.0739i)T + (0.781 + 0.623i)T^{2} \)
79 \( 1 - 1.56T + T^{2} \)
83 \( 1 + (0.781 + 0.623i)T^{2} \)
89 \( 1 + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (1.33 - 1.33i)T - iT^{2} \)
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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

−9.495422960862697847558404163352, −9.043910180536846686613734733673, −8.036458105745530535771878694333, −7.01075339824977007227570652815, −6.56308090249533826258671012189, −5.43533804143026284429264141679, −4.78740320317344805830414102141, −3.73722012311718243122796405131, −3.25511729515320450510884595474, −1.15830009665155743994143721128, 0.53706298022763699902901979500, 1.93129934096458834321183676159, 3.54321980091002795601970937762, 4.35054717158734382018347522774, 5.48279578370021835178889307553, 5.85967960245525969342072020334, 6.54310387678043141277599505377, 7.56734185222386629550312628621, 8.543511432016709389151769458293, 9.389988178124503376692326468657

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