Properties

Label 2-276-23.12-c1-0-1
Degree $2$
Conductor $276$
Sign $0.978 - 0.206i$
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  + (−0.142 + 0.989i)3-s + (3.91 − 1.15i)5-s + (−0.0825 − 0.0952i)7-s + (−0.959 − 0.281i)9-s + (−1.03 − 0.667i)11-s + (1.25 − 1.44i)13-s + (0.581 + 4.04i)15-s + (0.787 + 1.72i)17-s + (0.0613 − 0.134i)19-s + (0.106 − 0.0681i)21-s + (−2.15 + 4.28i)23-s + (9.83 − 6.31i)25-s + (0.415 − 0.909i)27-s + (3.11 + 6.82i)29-s + (0.335 + 2.33i)31-s + ⋯
L(s)  = 1  + (−0.0821 + 0.571i)3-s + (1.75 − 0.514i)5-s + (−0.0311 − 0.0359i)7-s + (−0.319 − 0.0939i)9-s + (−0.313 − 0.201i)11-s + (0.347 − 0.400i)13-s + (0.150 + 1.04i)15-s + (0.190 + 0.418i)17-s + (0.0140 − 0.0308i)19-s + (0.0231 − 0.0148i)21-s + (−0.448 + 0.893i)23-s + (1.96 − 1.26i)25-s + (0.0799 − 0.175i)27-s + (0.578 + 1.26i)29-s + (0.0602 + 0.419i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.978 - 0.206i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.978 - 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56371 + 0.163430i\)
\(L(\frac12)\) \(\approx\) \(1.56371 + 0.163430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (2.15 - 4.28i)T \)
good5 \( 1 + (-3.91 + 1.15i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.0825 + 0.0952i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (1.03 + 0.667i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-1.25 + 1.44i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-0.787 - 1.72i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.0613 + 0.134i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-3.11 - 6.82i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.335 - 2.33i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (7.41 + 2.17i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (10.0 - 2.95i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.71 + 11.9i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 5.71T + 47T^{2} \)
53 \( 1 + (4.37 + 5.04i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (4.98 - 5.75i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.180 + 1.25i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (8.94 - 5.75i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-7.77 + 4.99i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (5.25 - 11.4i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-2.44 + 2.81i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-7.82 - 2.29i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.00 + 6.98i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-12.7 + 3.73i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.97562144064522533571982812044, −10.51663852420488196382265258659, −10.19159538675990634112124721356, −9.130832280402774491839574698619, −8.417088909683587155941549421055, −6.73330999435376705734185587274, −5.62927267089568986221810171467, −5.07497521153060764648621000720, −3.34091520610292365253038656498, −1.71578711285566175275453896510, 1.75560068291132735179469174889, 2.85476536813044432320473613097, 4.90637258544649506170672908496, 6.12044546319307270417092378178, 6.59263382916107703156267637273, 7.916808474846965991987680048422, 9.156526032253820622406916633320, 9.985751294765142475689551173595, 10.76676055260101462262436650188, 11.93217828289917470576123054378

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