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Properties

Label 2-912-19.7-c1-0-15
Degree $2$
Conductor $912$
Sign $0.238 + 0.971i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.675 − 1.17i)5-s − 0.351·7-s + (−0.499 + 0.866i)9-s − 5.52·11-s + (2.58 − 4.47i)13-s + (0.675 − 1.17i)15-s + (2.43 − 3.61i)19-s + (−0.175 − 0.304i)21-s + (4.41 − 7.63i)23-s + (1.58 − 2.74i)25-s − 0.999·27-s + (−1.35 + 2.34i)29-s + 0.524·31-s + (−2.76 − 4.78i)33-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.302 − 0.523i)5-s − 0.133·7-s + (−0.166 + 0.288i)9-s − 1.66·11-s + (0.717 − 1.24i)13-s + (0.174 − 0.302i)15-s + (0.559 − 0.828i)19-s + (−0.0383 − 0.0665i)21-s + (0.919 − 1.59i)23-s + (0.317 − 0.549i)25-s − 0.192·27-s + (−0.251 + 0.434i)29-s + 0.0941·31-s + (−0.480 − 0.832i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.238 + 0.971i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941483 - 0.738285i\)
\(L(\frac12)\) \(\approx\) \(0.941483 - 0.738285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-2.43 + 3.61i)T \)
good5 \( 1 + (0.675 + 1.17i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.351T + 7T^{2} \)
11 \( 1 + 5.52T + 11T^{2} \)
13 \( 1 + (-2.58 + 4.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.41 + 7.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.35 - 2.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.524T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-1.35 - 2.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.26 + 5.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.02 + 3.51i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.76 + 4.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.938 + 1.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.99 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.52 - 4.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.85 + 6.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.91 - 6.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.34T + 83T^{2} \)
89 \( 1 + (-2.32 + 4.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.90 - 11.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.05377638928067847505445476805, −8.931683677950106157258207620673, −8.300475888816108055494294514740, −7.66333827151791883220836374355, −6.43680761381306210675429554561, −5.18091671986280360337340663577, −4.82035851860667695339351919157, −3.37616086036901105644933970646, −2.60874542782096559274396435377, −0.55164722450051100403758847716, 1.58567693003855994962986159818, 2.88744657702586208518653963793, 3.71050838011052802892397165134, 5.10059652804735857263724118839, 6.01325263220962409329849966776, 7.10077601123258952476627604911, 7.60323463006521486676244594224, 8.485459141860654492998577620945, 9.419974006724268869879423167624, 10.26013723112315403751006893547

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