Properties

Label 2-276-69.68-c1-0-1
Degree $2$
Conductor $276$
Sign $0.213 - 0.977i$
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  + (1.28 + 1.16i)3-s − 0.936·5-s + 3.88i·7-s + (0.280 + 2.98i)9-s − 4.27·11-s + 2.56·13-s + (−1.19 − 1.09i)15-s + 7.60·17-s − 6.07i·19-s + (−4.53 + 4.98i)21-s + (2.39 + 4.15i)23-s − 4.12·25-s + (−3.12 + 4.15i)27-s − 5.97i·29-s + 3.68·31-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)3-s − 0.418·5-s + 1.46i·7-s + (0.0935 + 0.995i)9-s − 1.28·11-s + 0.710·13-s + (−0.309 − 0.281i)15-s + 1.84·17-s − 1.39i·19-s + (−0.989 + 1.08i)21-s + (0.500 + 0.865i)23-s − 0.824·25-s + (−0.601 + 0.799i)27-s − 1.10i·29-s + 0.661·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.213 - 0.977i$
Analytic conductor: \(2.20387\)
Root analytic conductor: \(1.48454\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 276,\ (\ :1/2),\ 0.213 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13039 + 0.910425i\)
\(L(\frac12)\) \(\approx\) \(1.13039 + 0.910425i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 - 1.16i)T \)
23 \( 1 + (-2.39 - 4.15i)T \)
good5 \( 1 + 0.936T + 5T^{2} \)
7 \( 1 - 3.88iT - 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
17 \( 1 - 7.60T + 17T^{2} \)
19 \( 1 + 6.07iT - 19T^{2} \)
29 \( 1 + 5.97iT - 29T^{2} \)
31 \( 1 - 3.68T + 31T^{2} \)
37 \( 1 - 2.18iT - 37T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 - 6.07iT - 43T^{2} \)
47 \( 1 - 1.30iT - 47T^{2} \)
53 \( 1 - 8.13T + 53T^{2} \)
59 \( 1 + 4.66iT - 59T^{2} \)
61 \( 1 + 7.77iT - 61T^{2} \)
67 \( 1 + 11.6iT - 67T^{2} \)
71 \( 1 - 2.33iT - 71T^{2} \)
73 \( 1 + 5.68T + 73T^{2} \)
79 \( 1 - 1.70iT - 79T^{2} \)
83 \( 1 + 0.525T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 9.96iT - 97T^{2} \)
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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

−12.00255254706708102885420401805, −11.14721491516759808786925703332, −10.03803427076643348129315859025, −9.214646234199610983491166817733, −8.271773050427809451585667776041, −7.62232071269339407621228132160, −5.77057205288381172401598373208, −4.99429887855410325458190111892, −3.43429768865754316565780590496, −2.46519556850254437040356775606, 1.15459944877248117764411251767, 3.12225864317805090074164937788, 4.05276797566720825448117518440, 5.73296833877342311546368726279, 7.11359717857339011688054583829, 7.79676824116551012530971232427, 8.412229809525462587466596449058, 10.01956751287470982331840656759, 10.49918071287841470736730056694, 11.85713573266385480957527303495

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