Properties

Label 2-276-23.12-c1-0-0
Degree $2$
Conductor $276$
Sign $-0.511 - 0.859i$
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 + (−2.38 + 0.701i)5-s + (2.64 + 3.05i)7-s + (−0.959 − 0.281i)9-s + (−4.05 − 2.60i)11-s + (−2.81 + 3.24i)13-s + (−0.354 − 2.46i)15-s + (1.87 + 4.10i)17-s + (−1.84 + 4.02i)19-s + (−3.40 + 2.18i)21-s + (4.75 + 0.655i)23-s + (1.01 − 0.649i)25-s + (0.415 − 0.909i)27-s + (−0.207 − 0.454i)29-s + (−0.727 − 5.06i)31-s + ⋯
L(s)  = 1  + (−0.0821 + 0.571i)3-s + (−1.06 + 0.313i)5-s + (1.00 + 1.15i)7-s + (−0.319 − 0.0939i)9-s + (−1.22 − 0.786i)11-s + (−0.780 + 0.900i)13-s + (−0.0915 − 0.636i)15-s + (0.454 + 0.995i)17-s + (−0.422 + 0.924i)19-s + (−0.742 + 0.477i)21-s + (0.990 + 0.136i)23-s + (0.202 − 0.129i)25-s + (0.0799 − 0.175i)27-s + (−0.0385 − 0.0843i)29-s + (−0.130 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.511 - 0.859i$
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: \((2,\ 276,\ (\ :1/2),\ -0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433512 + 0.762126i\)
\(L(\frac12)\) \(\approx\) \(0.433512 + 0.762126i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.75 - 0.655i)T \)
good5 \( 1 + (2.38 - 0.701i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-2.64 - 3.05i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (4.05 + 2.60i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (2.81 - 3.24i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.87 - 4.10i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (1.84 - 4.02i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.207 + 0.454i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.727 + 5.06i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-10.2 - 3.00i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-7.29 + 2.14i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.07 + 7.50i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 7.67T + 47T^{2} \)
53 \( 1 + (-5.00 - 5.77i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (1.85 - 2.13i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.225 + 1.57i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-10.1 + 6.49i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-3.18 + 2.04i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (3.09 - 6.77i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-7.62 + 8.79i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (5.09 + 1.49i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (1.58 - 11.0i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (6.84 - 2.00i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.91977778709382989004299346402, −11.30675755265599431163585486695, −10.55512381343199660278442225735, −9.265894540126564324101741787347, −8.208864497476390860967296286926, −7.71567978733151905293709708314, −5.99096613161186216873481326976, −5.03706741992726713156607559833, −3.88790913042078681591874754374, −2.45679246091751529105620542149, 0.67945956578828975860742641776, 2.75626097715573329512361757947, 4.51784500522464605137860518830, 5.11489769407113224687177981392, 7.12885562925798068974688142039, 7.61044167782793486423450290829, 8.228539491974132772236102677714, 9.790344402835959133413844992765, 10.90091217038147404241161306052, 11.45583207634825233452146053257

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