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Properties

Label 2-912-57.41-c1-0-32
Degree $2$
Conductor $912$
Sign $0.642 + 0.766i$
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  + (1.67 − 0.453i)3-s + (1.96 + 0.346i)5-s + (−0.910 − 1.57i)7-s + (2.58 − 1.51i)9-s + (−4.10 − 2.37i)11-s + (0.151 − 0.415i)13-s + (3.44 − 0.312i)15-s + (1.07 − 1.28i)17-s + (3.58 − 2.48i)19-s + (−2.23 − 2.22i)21-s + (5.93 − 1.04i)23-s + (−0.952 − 0.346i)25-s + (3.63 − 3.71i)27-s + (4.91 − 4.12i)29-s + (−4.88 + 2.82i)31-s + ⋯
L(s)  = 1  + (0.965 − 0.262i)3-s + (0.879 + 0.155i)5-s + (−0.344 − 0.596i)7-s + (0.862 − 0.505i)9-s + (−1.23 − 0.715i)11-s + (0.0419 − 0.115i)13-s + (0.889 − 0.0808i)15-s + (0.260 − 0.310i)17-s + (0.821 − 0.569i)19-s + (−0.488 − 0.485i)21-s + (1.23 − 0.218i)23-s + (−0.190 − 0.0693i)25-s + (0.699 − 0.714i)27-s + (0.913 − 0.766i)29-s + (−0.877 + 0.506i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16939 - 1.01239i\)
\(L(\frac12)\) \(\approx\) \(2.16939 - 1.01239i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.453i)T \)
19 \( 1 + (-3.58 + 2.48i)T \)
good5 \( 1 + (-1.96 - 0.346i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.910 + 1.57i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.10 + 2.37i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.151 + 0.415i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.07 + 1.28i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-5.93 + 1.04i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-4.91 + 4.12i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (4.88 - 2.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 + (3.75 - 1.36i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.15 - 12.2i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.92 - 8.25i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.424 + 2.40i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.87 - 3.24i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.80 - 10.2i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.27 + 6.28i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.897 + 5.08i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (13.5 - 4.94i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.23 - 8.88i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.523 - 0.302i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.07 + 1.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.64 + 1.96i)T + (-16.8 - 95.5i)T^{2} \)
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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

−9.954084081499393438673425184884, −9.185691782264933765018287409460, −8.289623823318522471996033341626, −7.48309084587467971726391533404, −6.70287485512140147095298060079, −5.67710366808493724190030862193, −4.62183648325238097735648901847, −3.15562329253243522893943552844, −2.68060285951426047872816023632, −1.09874402161093035031552861382, 1.79591881507472220046756939139, 2.65778312874500696704864029603, 3.69130557981573875743193081446, 5.10902592753888205501841669828, 5.59760431794170168747301564542, 7.01521605032190297927033534933, 7.70075504093863481213261090886, 8.793126365930012563082158128308, 9.275397831083203243304812706536, 10.16259669230588158176641393627

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