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Properties

Label 2-912-19.12-c2-0-25
Degree $2$
Conductor $912$
Sign $0.334 + 0.942i$
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 + (−3.20 − 5.55i)5-s + 2.26·7-s + (1.5 − 2.59i)9-s + 20.0·11-s + (−0.135 − 0.0782i)13-s + (−9.62 − 5.55i)15-s + (12.3 + 21.3i)17-s + (18.9 + 1.60i)19-s + (3.39 − 1.95i)21-s + (2.62 − 4.54i)23-s + (−8.07 + 13.9i)25-s − 5.19i·27-s + (−31.4 − 18.1i)29-s + 17.1i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.641 − 1.11i)5-s + 0.323·7-s + (0.166 − 0.288i)9-s + 1.82·11-s + (−0.0104 − 0.00601i)13-s + (−0.641 − 0.370i)15-s + (0.724 + 1.25i)17-s + (0.996 + 0.0847i)19-s + (0.161 − 0.0933i)21-s + (0.114 − 0.197i)23-s + (−0.323 + 0.559i)25-s − 0.192i·27-s + (−1.08 − 0.626i)29-s + 0.551i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.334 + 0.942i$
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: \((2,\ 912,\ (\ :1),\ 0.334 + 0.942i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.402443112\)
\(L(\frac12)\) \(\approx\) \(2.402443112\)
\(L(2)\) 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 \( 1 + (-18.9 - 1.60i)T \)
good5 \( 1 + (3.20 + 5.55i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.26T + 49T^{2} \)
11 \( 1 - 20.0T + 121T^{2} \)
13 \( 1 + (0.135 + 0.0782i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.3 - 21.3i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-2.62 + 4.54i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (31.4 + 18.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 17.1iT - 961T^{2} \)
37 \( 1 + 42.7iT - 1.36e3T^{2} \)
41 \( 1 + (-30.0 + 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (12.5 + 21.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-14.6 + 25.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-48.4 - 27.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-29.9 + 17.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.3 + 47.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (66.0 + 38.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (63.6 - 36.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (45.9 + 79.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-53.1 + 30.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 148.T + 6.88e3T^{2} \)
89 \( 1 + (62.7 + 36.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (70.0 - 40.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.324679897063740506479284661687, −8.966274626985197829699296530550, −8.082492633407446799615856543719, −7.42456093933664782508670202079, −6.31521188870805874718483596951, −5.29586119075565701591739761457, −4.08754300956466918703298741470, −3.62529796242779277247770814157, −1.77726468133607707783078230001, −0.890632445643735087952025284014, 1.28766385106219709133154168023, 2.89805559422437532682659405679, 3.56182730401913329809326028371, 4.50740215561729105721878391394, 5.75639782156332840413160944878, 7.00960259092200673924660943698, 7.31827743031837835181650017347, 8.370869099228394829267838297413, 9.416045124756032532359318015788, 9.813250651449596898387265980800

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