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Properties

Label 2-26e2-52.47-c1-0-31
Degree $2$
Conductor $676$
Sign $0.736 + 0.676i$
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  + (−0.700 + 1.22i)2-s − 1.61i·3-s + (−1.01 − 1.72i)4-s + (−1.52 − 1.52i)5-s + (1.98 + 1.13i)6-s + (1.44 + 1.44i)7-s + (2.82 − 0.0454i)8-s + 0.386·9-s + (2.94 − 0.806i)10-s + (3.06 + 3.06i)11-s + (−2.78 + 1.64i)12-s + (−2.78 + 0.763i)14-s + (−2.47 + 2.47i)15-s + (−1.92 + 3.50i)16-s − 4.78i·17-s + (−0.270 + 0.474i)18-s + ⋯
L(s)  = 1  + (−0.495 + 0.868i)2-s − 0.933i·3-s + (−0.509 − 0.860i)4-s + (−0.683 − 0.683i)5-s + (0.810 + 0.462i)6-s + (0.546 + 0.546i)7-s + (0.999 − 0.0160i)8-s + 0.128·9-s + (0.932 − 0.255i)10-s + (0.923 + 0.923i)11-s + (−0.803 + 0.475i)12-s + (−0.745 + 0.203i)14-s + (−0.637 + 0.637i)15-s + (−0.481 + 0.876i)16-s − 1.16i·17-s + (−0.0638 + 0.111i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.736 + 0.676i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01262 - 0.394204i\)
\(L(\frac12)\) \(\approx\) \(1.01262 - 0.394204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.700 - 1.22i)T \)
13 \( 1 \)
good3 \( 1 + 1.61iT - 3T^{2} \)
5 \( 1 + (1.52 + 1.52i)T + 5iT^{2} \)
7 \( 1 + (-1.44 - 1.44i)T + 7iT^{2} \)
11 \( 1 + (-3.06 - 3.06i)T + 11iT^{2} \)
17 \( 1 + 4.78iT - 17T^{2} \)
19 \( 1 + (-1.64 + 1.64i)T - 19iT^{2} \)
23 \( 1 + 4.91T + 23T^{2} \)
29 \( 1 - 5.88T + 29T^{2} \)
31 \( 1 + (-0.420 + 0.420i)T - 31iT^{2} \)
37 \( 1 + (-1.36 + 1.36i)T - 37iT^{2} \)
41 \( 1 + (-1.09 - 1.09i)T + 41iT^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (8.07 + 8.07i)T + 47iT^{2} \)
53 \( 1 + 1.33T + 53T^{2} \)
59 \( 1 + (4.74 + 4.74i)T + 59iT^{2} \)
61 \( 1 + 0.717T + 61T^{2} \)
67 \( 1 + (5.00 - 5.00i)T - 67iT^{2} \)
71 \( 1 + (1.24 - 1.24i)T - 71iT^{2} \)
73 \( 1 + (-5.35 + 5.35i)T - 73iT^{2} \)
79 \( 1 - 1.11iT - 79T^{2} \)
83 \( 1 + (2.45 - 2.45i)T - 83iT^{2} \)
89 \( 1 + (1.37 - 1.37i)T - 89iT^{2} \)
97 \( 1 + (-0.971 - 0.971i)T + 97iT^{2} \)
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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

−10.06198036018816784439508005934, −9.282937854594739217658891635292, −8.443107011205862005965912169778, −7.71860604496894715659502904933, −7.06430561257825413778325548378, −6.20939127304583190695640695340, −4.95194632024679823338462953887, −4.28828366787961257651975256509, −2.03741951344865930209319168744, −0.821264524461570371254223485546, 1.35484551570614929969876627504, 3.17879322856334368967218686416, 3.92659714562331919201067237354, 4.51058305587084404058226315323, 6.15658357302611285884544848783, 7.44431370957072124646533025115, 8.139728220883195888716219751070, 9.060338821172853031573751098868, 9.962008428748277415397539574636, 10.69303669743156604184328516365

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