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Properties

Label 2-26e2-13.5-c2-0-11
Degree $2$
Conductor $676$
Sign $0.801 + 0.597i$
Analytic cond. $18.4196$
Root an. cond. $4.29181$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.17·3-s + (−1.94 + 1.94i)5-s + (1.43 + 1.43i)7-s − 7.62·9-s + (−4.89 − 4.89i)11-s + (2.28 − 2.28i)15-s + 8.60i·17-s + (18.4 − 18.4i)19-s + (−1.68 − 1.68i)21-s − 23.4i·23-s + 17.4i·25-s + 19.5·27-s + 31.9·29-s + (−6.90 + 6.90i)31-s + (5.74 + 5.74i)33-s + ⋯
L(s)  = 1  − 0.391·3-s + (−0.389 + 0.389i)5-s + (0.205 + 0.205i)7-s − 0.847·9-s + (−0.444 − 0.444i)11-s + (0.152 − 0.152i)15-s + 0.506i·17-s + (0.973 − 0.973i)19-s + (−0.0802 − 0.0802i)21-s − 1.01i·23-s + 0.697i·25-s + 0.722·27-s + 1.10·29-s + (−0.222 + 0.222i)31-s + (0.173 + 0.173i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.801 + 0.597i$
Analytic conductor: \(18.4196\)
Root analytic conductor: \(4.29181\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1),\ 0.801 + 0.597i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.145389028\)
\(L(\frac12)\) \(\approx\) \(1.145389028\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + 1.17T + 9T^{2} \)
5 \( 1 + (1.94 - 1.94i)T - 25iT^{2} \)
7 \( 1 + (-1.43 - 1.43i)T + 49iT^{2} \)
11 \( 1 + (4.89 + 4.89i)T + 121iT^{2} \)
17 \( 1 - 8.60iT - 289T^{2} \)
19 \( 1 + (-18.4 + 18.4i)T - 361iT^{2} \)
23 \( 1 + 23.4iT - 529T^{2} \)
29 \( 1 - 31.9T + 841T^{2} \)
31 \( 1 + (6.90 - 6.90i)T - 961iT^{2} \)
37 \( 1 + (30.6 + 30.6i)T + 1.36e3iT^{2} \)
41 \( 1 + (-52.3 + 52.3i)T - 1.68e3iT^{2} \)
43 \( 1 - 74.1iT - 1.84e3T^{2} \)
47 \( 1 + (-4.77 - 4.77i)T + 2.20e3iT^{2} \)
53 \( 1 - 72.4T + 2.80e3T^{2} \)
59 \( 1 + (-16.2 - 16.2i)T + 3.48e3iT^{2} \)
61 \( 1 + 8.63T + 3.72e3T^{2} \)
67 \( 1 + (-68.1 + 68.1i)T - 4.48e3iT^{2} \)
71 \( 1 + (-77.4 + 77.4i)T - 5.04e3iT^{2} \)
73 \( 1 + (55.0 + 55.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 49.9T + 6.24e3T^{2} \)
83 \( 1 + (-66.1 + 66.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (78.9 + 78.9i)T + 7.92e3iT^{2} \)
97 \( 1 + (-43.9 + 43.9i)T - 9.40e3iT^{2} \)
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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

−10.54231200190666119493781583429, −9.218211618251764022533593102917, −8.492017153597496365056706360666, −7.58623196283336710426599218606, −6.61400513389798328493807462207, −5.65827948553033643605668756567, −4.85236187507695217445749446833, −3.46217411679193368184287381563, −2.49953857588025506867517608030, −0.56804580413508321780289774083, 0.956826396039489747927836915056, 2.63828139195851170039845913230, 3.89566401599312043333432601525, 5.04213482888584949549906523777, 5.68894831556424972915128930050, 6.90247950049744666828514349026, 7.85351483760064353615813307908, 8.507876908754794885223353355236, 9.618155042814922927417208231408, 10.37888902404182003391833277320

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