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Properties

Label 2-276-69.68-c1-0-3
Degree $2$
Conductor $276$
Sign $0.952 - 0.303i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.952 - 0.303i$
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: \((2,\ 276,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70648 + 0.265309i\)
\(L(\frac12)\) \(\approx\) \(1.70648 + 0.265309i\)
\(L(\frac{3}{2})\) 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

−11.72762754609030137117543927248, −10.77208399979737025312246486785, −10.00830401634952656613006102529, −9.186978888958113917380772517965, −8.191894868592916541403787362380, −7.10091331106449351953578345890, −5.96412810965949731480821721374, −4.16008693883498986071377394083, −3.84869294200729240636375506270, −1.82047210164072324471187938083, 1.81830820278618611233483936579, 2.91571866804644427937028347328, 4.54085846459526942227401791312, 6.28353490374833779956929245149, 6.58139213457258989418136556714, 8.267341432937453420498883858498, 8.982572745768847606139273311872, 9.439533731828760013032705522376, 11.17802655489572389623569489096, 11.89366359705217870086563473347

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