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Properties

Label 2-912-57.29-c1-0-2
Degree $2$
Conductor $912$
Sign $-0.676 - 0.736i$
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  + (0.716 − 1.57i)3-s + (−1.14 + 3.13i)5-s + (1.07 − 1.85i)7-s + (−1.97 − 2.25i)9-s + (−5.41 + 3.12i)11-s + (−2.56 + 3.05i)13-s + (4.12 + 4.04i)15-s + (0.403 − 0.0711i)17-s + (−4.34 − 0.329i)19-s + (−2.16 − 3.02i)21-s + (−0.280 − 0.770i)23-s + (−4.70 − 3.94i)25-s + (−4.97 + 1.49i)27-s + (−0.805 + 4.56i)29-s + (2.02 + 1.16i)31-s + ⋯
L(s)  = 1  + (0.413 − 0.910i)3-s + (−0.510 + 1.40i)5-s + (0.405 − 0.702i)7-s + (−0.657 − 0.753i)9-s + (−1.63 + 0.943i)11-s + (−0.710 + 0.846i)13-s + (1.06 + 1.04i)15-s + (0.0978 − 0.0172i)17-s + (−0.997 − 0.0756i)19-s + (−0.472 − 0.660i)21-s + (−0.0584 − 0.160i)23-s + (−0.940 − 0.788i)25-s + (−0.957 + 0.287i)27-s + (−0.149 + 0.847i)29-s + (0.363 + 0.210i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.676 - 0.736i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.676 - 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181814 + 0.413736i\)
\(L(\frac12)\) \(\approx\) \(0.181814 + 0.413736i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.716 + 1.57i)T \)
19 \( 1 + (4.34 + 0.329i)T \)
good5 \( 1 + (1.14 - 3.13i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.07 + 1.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.41 - 3.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.56 - 3.05i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.403 + 0.0711i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.280 + 0.770i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.805 - 4.56i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.02 - 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.01iT - 37T^{2} \)
41 \( 1 + (0.926 - 0.777i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (5.87 + 2.13i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (7.59 + 1.33i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.220 + 0.0802i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.930 - 5.27i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.30 + 2.65i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (3.48 + 0.614i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-4.19 - 1.52i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (4.33 - 3.63i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-8.05 - 9.59i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.01 - 4.62i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.61 - 4.71i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (16.0 - 2.83i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.54667404077325069942154359014, −9.704250786910144105160252327981, −8.388259228727094874124796563583, −7.65438992248421779791880568360, −7.09972078480832456306269901713, −6.59663695387064080651739174360, −5.11804954745025204140408549012, −3.96222880984521614248418112528, −2.77619082543514151132603087303, −2.03028748500752183245682281316, 0.18472637067323247548721602027, 2.31377754531117763002047942382, 3.36776698789068655636002620443, 4.69767988645045423955090181812, 5.09738228468003520963560533222, 5.89153438547009577099685681552, 7.909046362535886611892712010145, 8.173798569415283343665567120141, 8.741501756259723744297615721964, 9.776449133210900535844894648751

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