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Properties

Label 2-26e2-52.15-c1-0-49
Degree $2$
Conductor $676$
Sign $0.999 - 0.0386i$
Analytic cond. $5.39788$
Root an. cond. $2.32333$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (3 − 3i)5-s + (1.99 + 2i)8-s + (−1.5 + 2.59i)9-s + (5.19 − 3i)10-s + (1.99 + 3.46i)16-s + (−1.73 − i)17-s + (−3 + 3i)18-s + (8.19 − 2.19i)20-s − 13i·25-s + (−2 − 3.46i)29-s + (1.46 + 5.46i)32-s + (−1.99 − 2i)34-s + (−5.19 + 3i)36-s + (−1.83 + 6.83i)37-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + (1.34 − 1.34i)5-s + (0.707 + 0.707i)8-s + (−0.5 + 0.866i)9-s + (1.64 − 0.948i)10-s + (0.499 + 0.866i)16-s + (−0.420 − 0.242i)17-s + (−0.707 + 0.707i)18-s + (1.83 − 0.491i)20-s − 2.60i·25-s + (−0.371 − 0.643i)29-s + (0.258 + 0.965i)32-s + (−0.342 − 0.342i)34-s + (−0.866 + 0.5i)36-s + (−0.300 + 1.12i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.999 - 0.0386i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.999 - 0.0386i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.23378 + 0.0624420i\)
\(L(\frac12)\) \(\approx\) \(3.23378 + 0.0624420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
13 \( 1 \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3 + 3i)T - 5iT^{2} \)
7 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (1.83 - 6.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.36 + 0.366i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11 - 11i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (1.09 - 4.09i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.83 + 6.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.56816515940329436587741006307, −9.629824856616633219605656292669, −8.649021013427562390158414370461, −7.950561825379673843031600399636, −6.63797318787420513637596818269, −5.72170782687909150393604702643, −5.13789183619840115935627461882, −4.35224066470441988035796298028, −2.67190764120389758005385419319, −1.68592757397874732013627925692, 1.82731276170867514234143650167, 2.85564073930611767716720073002, 3.66817314207737757395427194397, 5.18823651061258684904304454003, 6.10908377100309000693418573652, 6.52685859983827212587737400614, 7.45490883055894980511526779478, 9.119111432884457291585416254660, 9.808167447534433768546796658321, 10.78178213230309285380126183828

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