Properties

Label 2-912-57.14-c1-0-8
Degree $2$
Conductor $912$
Sign $0.174 - 0.984i$
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  + (−1.08 + 1.34i)3-s + (0.343 − 0.408i)5-s + (0.716 + 1.24i)7-s + (−0.637 − 2.93i)9-s + (−1.25 − 0.725i)11-s + (2.94 − 0.519i)13-s + (0.178 + 0.907i)15-s + (1.89 + 5.20i)17-s + (4.35 − 0.143i)19-s + (−2.45 − 0.382i)21-s + (−0.396 − 0.472i)23-s + (0.818 + 4.64i)25-s + (4.64 + 2.32i)27-s + (−4.97 − 1.81i)29-s + (4.28 − 2.47i)31-s + ⋯
L(s)  = 1  + (−0.627 + 0.778i)3-s + (0.153 − 0.182i)5-s + (0.270 + 0.469i)7-s + (−0.212 − 0.977i)9-s + (−0.378 − 0.218i)11-s + (0.817 − 0.144i)13-s + (0.0461 + 0.234i)15-s + (0.459 + 1.26i)17-s + (0.999 − 0.0329i)19-s + (−0.535 − 0.0834i)21-s + (−0.0826 − 0.0985i)23-s + (0.163 + 0.928i)25-s + (0.894 + 0.447i)27-s + (−0.924 − 0.336i)29-s + (0.769 − 0.444i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.174 - 0.984i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985469 + 0.825789i\)
\(L(\frac12)\) \(\approx\) \(0.985469 + 0.825789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 1.34i)T \)
19 \( 1 + (-4.35 + 0.143i)T \)
good5 \( 1 + (-0.343 + 0.408i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.716 - 1.24i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.25 + 0.725i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 0.519i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.89 - 5.20i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (0.396 + 0.472i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.97 + 1.81i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.28 + 2.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.41iT - 37T^{2} \)
41 \( 1 + (1.37 - 7.78i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (4.88 + 4.10i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (4.37 - 12.0i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (1.41 - 1.18i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-1.75 + 0.639i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-9.02 + 7.57i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.17 - 8.71i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-9.59 - 8.05i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.80 - 15.9i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-7.87 - 1.38i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.29 - 2.48i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.832 + 4.71i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.83 + 7.79i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.16082963911223642239874283649, −9.660881013414535782029914090615, −8.621023041408215281931613989102, −7.986150897216160017099790157772, −6.59674845432394111385862914556, −5.73090916779944206405584877980, −5.19900171515059339975423689792, −4.02622684695898229881674375759, −3.09232165598893987832146033164, −1.31328403087421150784817587978, 0.76098104488182925147986088423, 2.08563109522047696657598577059, 3.41471703604314661391987813304, 4.83963682663203660025651377647, 5.54535610058174849469040491658, 6.57627079571627584813198581538, 7.32139600987189956603325131607, 7.962064831133746957808717963244, 9.039979580501124176237563248605, 10.12360781126425838541245165567

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