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Properties

Label 2-912-19.12-c2-0-39
Degree $2$
Conductor $912$
Sign $-0.824 - 0.565i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + (−4.80 − 8.31i)5-s + 8.01·7-s + (1.5 − 2.59i)9-s − 18.0·11-s + (−7.29 − 4.21i)13-s + (−14.4 − 8.31i)15-s + (−2.70 − 4.69i)17-s + (−16.2 + 9.82i)19-s + (12.0 − 6.94i)21-s + (5.88 − 10.1i)23-s + (−33.5 + 58.1i)25-s − 5.19i·27-s + (19.7 + 11.3i)29-s + 34.5i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.960 − 1.66i)5-s + 1.14·7-s + (0.166 − 0.288i)9-s − 1.64·11-s + (−0.561 − 0.324i)13-s + (−0.960 − 0.554i)15-s + (−0.159 − 0.275i)17-s + (−0.855 + 0.517i)19-s + (0.572 − 0.330i)21-s + (0.255 − 0.442i)23-s + (−1.34 + 2.32i)25-s − 0.192i·27-s + (0.679 + 0.392i)29-s + 1.11i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.565i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.824 - 0.565i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ -0.824 - 0.565i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5639232468\)
\(L(\frac12)\) \(\approx\) \(0.5639232468\)
\(L(2)\) 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 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (16.2 - 9.82i)T \)
good5 \( 1 + (4.80 + 8.31i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 8.01T + 49T^{2} \)
11 \( 1 + 18.0T + 121T^{2} \)
13 \( 1 + (7.29 + 4.21i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (2.70 + 4.69i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-5.88 + 10.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-19.7 - 11.3i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 34.5iT - 961T^{2} \)
37 \( 1 + 16.2iT - 1.36e3T^{2} \)
41 \( 1 + (-66.4 + 38.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.88 + 4.98i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (17.0 - 29.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (19.8 + 11.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-29.3 + 16.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.8 - 70.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.87 + 5.70i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (58.5 - 33.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (3.21 + 5.56i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-12.6 + 7.31i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 126.T + 6.88e3T^{2} \)
89 \( 1 + (113. + 65.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-60.9 + 35.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.004223451519206811090773671689, −8.432911940674810472576369974520, −7.901213680612974629849958735428, −7.31118323335062748359528327570, −5.54281743133192418939818890784, −4.85624195728729586340665601814, −4.22388943329157607723575860641, −2.71603174075367194253380947396, −1.43687881928623517306179341144, −0.16643643563064882354321982195, 2.29802123475378804405763216289, 2.87424868109663423435824451606, 4.13274902485463545667407920650, 4.86791580801614906932904221038, 6.23530009070812149658342625314, 7.32554447657655829965073979606, 7.83193086017173834033223886208, 8.378917819957992786890748934763, 9.772227340925101557096971532915, 10.54324052394836727928590199956

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