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Properties

Label 2-912-19.12-c2-0-19
Degree $2$
Conductor $912$
Sign $0.976 + 0.214i$
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 + (−1.60 − 2.77i)5-s + 13.4·7-s + (1.5 − 2.59i)9-s + 3.80·11-s + (10.8 + 6.27i)13-s + (−4.81 − 2.77i)15-s + (14.4 + 25.0i)17-s + (−11.3 + 15.2i)19-s + (20.1 − 11.6i)21-s + (−14.1 + 24.5i)23-s + (7.35 − 12.7i)25-s − 5.19i·27-s + (23.9 + 13.8i)29-s − 40.4i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.320 − 0.555i)5-s + 1.92·7-s + (0.166 − 0.288i)9-s + 0.345·11-s + (0.835 + 0.482i)13-s + (−0.320 − 0.185i)15-s + (0.851 + 1.47i)17-s + (−0.598 + 0.800i)19-s + (0.960 − 0.554i)21-s + (−0.615 + 1.06i)23-s + (0.294 − 0.509i)25-s − 0.192i·27-s + (0.825 + 0.476i)29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.976 + 0.214i$
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: \((2,\ 912,\ (\ :1),\ 0.976 + 0.214i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.919234900\)
\(L(\frac12)\) \(\approx\) \(2.919234900\)
\(L(2)\) 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 \( 1 + (11.3 - 15.2i)T \)
good5 \( 1 + (1.60 + 2.77i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 13.4T + 49T^{2} \)
11 \( 1 - 3.80T + 121T^{2} \)
13 \( 1 + (-10.8 - 6.27i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-14.4 - 25.0i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (14.1 - 24.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-23.9 - 13.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 40.4iT - 961T^{2} \)
37 \( 1 + 27.0iT - 1.36e3T^{2} \)
41 \( 1 + (54.3 - 31.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (16.4 + 28.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.9 - 69.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-14.5 - 8.41i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-72.4 + 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35.8 + 62.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (81.8 + 47.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-46.5 + 26.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-51.3 - 89.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (67.8 - 39.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 4.94T + 6.88e3T^{2} \)
89 \( 1 + (22.6 + 13.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-132. + 76.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.826172356312744782922942193714, −8.587403803419556746667366521118, −8.263386500323299338098887298545, −7.74557089098741284874806245009, −6.41258925406038562241663095816, −5.45419493661071576144432607931, −4.34631882379647346305402026183, −3.72503400668567289451090227842, −1.86041231899687768583263633720, −1.31900128598308904074095555251, 1.12277907407597163292760565453, 2.44758130294499318346622532945, 3.51922968637238335854798102433, 4.65970796675148635314307281259, 5.24736433053511523863076844424, 6.71334600457605140094160223012, 7.49302302362234465793901660017, 8.476607495528291511663722786189, 8.671474624558165550401191908573, 10.15329890936951593853491906802

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