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Properties

Label 2-912-19.17-c1-0-16
Degree $2$
Conductor $912$
Sign $0.854 + 0.519i$
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  + (0.766 + 0.642i)3-s + (3.20 − 1.16i)5-s + (−1.43 − 2.49i)7-s + (0.173 + 0.984i)9-s + (−0.173 + 0.300i)11-s + (−1.26 + 1.06i)13-s + (3.20 + 1.16i)15-s + (1.20 − 6.83i)17-s + (2.82 − 3.31i)19-s + (0.500 − 2.83i)21-s + (6.39 + 2.32i)23-s + (5.08 − 4.26i)25-s + (−0.500 + 0.866i)27-s + (1.10 + 6.25i)29-s + (0.798 + 1.38i)31-s + ⋯
L(s)  = 1  + (0.442 + 0.371i)3-s + (1.43 − 0.521i)5-s + (−0.544 − 0.942i)7-s + (0.0578 + 0.328i)9-s + (−0.0523 + 0.0906i)11-s + (−0.351 + 0.294i)13-s + (0.827 + 0.301i)15-s + (0.292 − 1.65i)17-s + (0.648 − 0.761i)19-s + (0.109 − 0.618i)21-s + (1.33 + 0.484i)23-s + (1.01 − 0.853i)25-s + (−0.0962 + 0.166i)27-s + (0.204 + 1.16i)29-s + (0.143 + 0.248i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.854 + 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13431 - 0.598468i\)
\(L(\frac12)\) \(\approx\) \(2.13431 - 0.598468i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-2.82 + 3.31i)T \)
good5 \( 1 + (-3.20 + 1.16i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.43 + 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.26 - 1.06i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.20 + 6.83i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-6.39 - 2.32i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.10 - 6.25i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.798 - 1.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + (2.67 + 2.24i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.14 + 0.780i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.971 + 5.51i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (1.86 + 0.677i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.0773 - 0.438i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-11.7 - 4.28i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.187 - 1.06i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (15.6 - 5.68i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-9.51 - 7.98i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-8.36 - 7.02i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-5.85 - 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.37 + 1.15i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.634 + 3.59i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.884992226634945096710460414892, −9.294931415928254051018939433607, −8.724520040812808692155603067637, −7.11240017665034013858180436653, −6.94367889096208036666472932170, −5.24281585825146109694561140762, −5.04442535908172923999168695749, −3.51761039911733996323661124083, −2.53965684297629802787340502179, −1.10194035797027313256346712363, 1.65877458267209226100391789598, 2.58863457939569923675078311972, 3.45078067921121123017215990441, 5.17185111262734469171034311758, 6.06322426153417502958103735697, 6.45151475870535825102744135379, 7.66854237898535540541020139012, 8.620730149970394350592167151585, 9.364767229277637071762747060947, 10.08239747850736511663948037412

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