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Properties

Label 2-26e2-52.11-c1-0-21
Degree $2$
Conductor $676$
Sign $-0.852 - 0.522i$
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  + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + (3 + 3i)5-s + (2 − 1.99i)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 + (−2.19 − 8.19i)20-s + 13i·25-s + (−2 − 3.46i)29-s + (−5.46 + 1.46i)32-s + (−2 + 1.99i)34-s + (5.19 − 3i)36-s + (6.83 + 1.83i)37-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)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 + (−0.491 − 1.83i)20-s + 2.60i·25-s + (−0.371 − 0.643i)29-s + (−0.965 + 0.258i)32-s + (−0.342 + 0.342i)34-s + (0.866 − 0.5i)36-s + (1.12 + 0.300i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.852 - 0.522i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ -0.852 - 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373729 + 1.32613i\)
\(L(\frac12)\) \(\approx\) \(0.373729 + 1.32613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
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 \( 1 + (-6.83 - 1.83i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.366 + 1.36i)T + (-35.5 - 20.5i)T^{2} \)
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 \( 1 + (-4.09 - 1.09i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.83 + 1.83i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.60979845048149263333218269247, −9.897286603694313186915734159965, −9.231748434323597644927826497378, −8.030141007886339660248448996134, −7.32650536963385638499594939900, −6.24889838888420217726863325336, −5.86255091845933712618838626348, −4.79414530924544780636651368205, −3.15956544893362667504043846141, −1.90318312503936632538056224422, 0.832882557598125312580873013832, 1.96469539602603035875951488532, 3.26304847748242177882382049717, 4.59826070251664596999830374315, 5.40715543624795209968474517476, 6.32697496785207348441542333847, 7.947445050499528633989426308714, 8.790726349946872776840513089363, 9.450544690426406982458687399905, 9.836767175023846855540499655496

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