Properties

Label 2-276-23.18-c1-0-2
Degree $2$
Conductor $276$
Sign $0.383 + 0.923i$
Analytic cond. $2.20387$
Root an. cond. $1.48454$
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  + (−0.654 + 0.755i)3-s + (0.217 − 1.51i)5-s + (−1.55 − 3.40i)7-s + (−0.142 − 0.989i)9-s + (1.04 − 0.306i)11-s + (1.25 − 2.75i)13-s + (0.999 + 1.15i)15-s + (2.61 − 1.67i)17-s + (−3.49 − 2.24i)19-s + (3.59 + 1.05i)21-s + (2.35 + 4.17i)23-s + (2.56 + 0.752i)25-s + (0.841 + 0.540i)27-s + (2.24 − 1.44i)29-s + (−4.45 − 5.14i)31-s + ⋯
L(s)  = 1  + (−0.378 + 0.436i)3-s + (0.0971 − 0.675i)5-s + (−0.588 − 1.28i)7-s + (−0.0474 − 0.329i)9-s + (0.314 − 0.0924i)11-s + (0.348 − 0.763i)13-s + (0.258 + 0.297i)15-s + (0.633 − 0.406i)17-s + (−0.801 − 0.514i)19-s + (0.784 + 0.230i)21-s + (0.490 + 0.871i)23-s + (0.512 + 0.150i)25-s + (0.161 + 0.104i)27-s + (0.416 − 0.267i)29-s + (−0.800 − 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.383 + 0.923i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.383 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844931 - 0.564303i\)
\(L(\frac12)\) \(\approx\) \(0.844931 - 0.564303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-2.35 - 4.17i)T \)
good5 \( 1 + (-0.217 + 1.51i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (1.55 + 3.40i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-1.04 + 0.306i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.25 + 2.75i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-2.61 + 1.67i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (3.49 + 2.24i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-2.24 + 1.44i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.45 + 5.14i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.973 + 6.77i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.41 - 9.85i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.62 - 3.02i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.56T + 47T^{2} \)
53 \( 1 + (1.19 + 2.61i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (6.11 - 13.3i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-3.42 - 3.95i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-4.44 - 1.30i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-14.6 - 4.29i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-2.70 - 1.73i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (1.28 - 2.81i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (0.773 + 5.38i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-9.86 + 11.3i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-0.546 + 3.79i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.55888613637923699589074201941, −10.71848031150725793680178702685, −9.873439618699385232466704211674, −9.036975281482694232316690030518, −7.78206979281857942058673731699, −6.71879737109821401466870865570, −5.55329969739753103366606864152, −4.42320773213047918543920054179, −3.35679416204171119199026360836, −0.853290761524924203036743056111, 2.04343498870579159322690080433, 3.40248283075114421657503639284, 5.13567184016359121321905549635, 6.35311559213067962005021197255, 6.74855788688617894579786066020, 8.313389210831905279593275706341, 9.123699360693503082081489988281, 10.31472786000520518068121516855, 11.14289667133828087845626589694, 12.36948760051108427296643825222

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