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Properties

Label 2-912-19.12-c2-0-14
Degree $2$
Conductor $912$
Sign $0.412 + 0.910i$
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 + 1.73i)5-s − 11·7-s + (1.5 − 2.59i)9-s + 8·11-s + (−4.5 − 2.59i)13-s + (−3 − 1.73i)15-s + (13 + 22.5i)17-s − 19·19-s + (16.5 − 9.52i)21-s + (−16 + 27.7i)23-s + (10.5 − 18.1i)25-s + 5.19i·27-s + (−21 − 12.1i)29-s − 53.6i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.200 + 0.346i)5-s − 1.57·7-s + (0.166 − 0.288i)9-s + 0.727·11-s + (−0.346 − 0.199i)13-s + (−0.200 − 0.115i)15-s + (0.764 + 1.32i)17-s − 19-s + (0.785 − 0.453i)21-s + (−0.695 + 1.20i)23-s + (0.419 − 0.727i)25-s + 0.192i·27-s + (−0.724 − 0.418i)29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.412 + 0.910i$
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: \((2,\ 912,\ (\ :1),\ 0.412 + 0.910i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7651286900\)
\(L(\frac12)\) \(\approx\) \(0.7651286900\)
\(L(2)\) 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 \( 1 + 19T \)
good5 \( 1 + (-1 - 1.73i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 11T + 49T^{2} \)
11 \( 1 - 8T + 121T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-13 - 22.5i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (16 - 27.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (21 + 12.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 53.6iT - 961T^{2} \)
37 \( 1 + 46.7iT - 1.36e3T^{2} \)
41 \( 1 + (12 - 6.92i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-23.5 - 40.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-35 + 60.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (6 + 3.46i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-93 + 53.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (17.5 - 30.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.5 + 7.79i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-114 + 65.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (29.5 + 51.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-22.5 + 12.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 2T + 6.88e3T^{2} \)
89 \( 1 + (69 + 39.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (18 - 10.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.826193117897481048703634304477, −9.228323256304648841787987717975, −8.021596726805085442830975814789, −6.96876332075509595668309230635, −6.10539111789378810591762847352, −5.77549358539835775961093168791, −4.09572393445894342387781764718, −3.54911642293396019888450835512, −2.14426243533029911295299311466, −0.31981156867561010391259947647, 0.989809644471434917879001233345, 2.57038724343422322662046559313, 3.67711112334076759788917204580, 4.85701710422782470659853411768, 5.81781026168899803235473350410, 6.72538149481743279217537647881, 7.11647464851411437699259291730, 8.547901562177282876531661989983, 9.304943289029110078747822570104, 9.982506531401202940433795749999

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