Properties

Label 2-26e2-52.19-c1-0-44
Degree $2$
Conductor $676$
Sign $0.999 - 0.0257i$
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 + (2.36 − 2.36i)5-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + (4.09 + 2.36i)10-s + (1.99 − 3.46i)16-s + (5.13 − 2.96i)17-s + (3 − 3i)18-s + (−1.73 + 6.46i)20-s − 6.19i·25-s + (5.33 − 9.23i)29-s + (5.46 + 1.46i)32-s + (5.92 + 5.92i)34-s + (5.19 + 3i)36-s + (−4.86 + 1.30i)37-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (1.05 − 1.05i)5-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + (1.29 + 0.748i)10-s + (0.499 − 0.866i)16-s + (1.24 − 0.718i)17-s + (0.707 − 0.707i)18-s + (−0.387 + 1.44i)20-s − 1.23i·25-s + (0.989 − 1.71i)29-s + (0.965 + 0.258i)32-s + (1.01 + 1.01i)34-s + (0.866 + 0.5i)36-s + (−0.799 + 0.214i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.999 - 0.0257i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.999 - 0.0257i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73919 + 0.0224282i\)
\(L(\frac12)\) \(\approx\) \(1.73919 + 0.0224282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 + 2.36i)T - 5iT^{2} \)
7 \( 1 + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-5.13 + 2.96i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.33 + 9.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (4.86 - 1.30i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.03 - 11.3i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 3.53T + 53T^{2} \)
59 \( 1 + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-7.69 - 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.16 - 1.16i)T + 73iT^{2} \)
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

−9.891194655416715307731638869614, −9.619060448833484646184301856370, −8.669826976901062253771480457483, −8.007557871575720396519201258417, −6.75271598306140148196699543430, −5.89705107854713829278951519308, −5.30911495426122383069925230895, −4.29663977408520249716193664816, −2.93930545219904912563539654278, −0.929592247353656641074499146664, 1.68275257327641620691585412187, 2.68518357382143102989490540937, 3.56751740183062443398677899422, 5.12908325914177984316994507770, 5.73134097861078263029937945433, 6.78568012093346570530914999428, 8.057908086287600414248586003869, 9.020416028165239354782123811946, 10.02174827988697067445134734297, 10.55341682461199103790928990983

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