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Properties

Label 2-26e2-52.31-c1-0-29
Degree $2$
Conductor $676$
Sign $0.662 + 0.749i$
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  + (−0.492 − 1.32i)2-s − 0.850i·3-s + (−1.51 + 1.30i)4-s + (0.166 − 0.166i)5-s + (−1.12 + 0.418i)6-s + (−1.86 + 1.86i)7-s + (2.47 + 1.36i)8-s + 2.27·9-s + (−0.302 − 0.138i)10-s + (−1.01 + 1.01i)11-s + (1.10 + 1.28i)12-s + (3.39 + 1.55i)14-s + (−0.141 − 0.141i)15-s + (0.593 − 3.95i)16-s − 1.39i·17-s + (−1.12 − 3.01i)18-s + ⋯
L(s)  = 1  + (−0.348 − 0.937i)2-s − 0.490i·3-s + (−0.757 + 0.652i)4-s + (0.0744 − 0.0744i)5-s + (−0.460 + 0.170i)6-s + (−0.706 + 0.706i)7-s + (0.875 + 0.483i)8-s + 0.759·9-s + (−0.0956 − 0.0438i)10-s + (−0.307 + 0.307i)11-s + (0.320 + 0.371i)12-s + (0.908 + 0.416i)14-s + (−0.0365 − 0.0365i)15-s + (0.148 − 0.988i)16-s − 0.339i·17-s + (−0.264 − 0.711i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(5.39788\)
Root analytic conductor: \(2.32333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06082 - 0.478115i\)
\(L(\frac12)\) \(\approx\) \(1.06082 - 0.478115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.492 + 1.32i)T \)
13 \( 1 \)
good3 \( 1 + 0.850iT - 3T^{2} \)
5 \( 1 + (-0.166 + 0.166i)T - 5iT^{2} \)
7 \( 1 + (1.86 - 1.86i)T - 7iT^{2} \)
11 \( 1 + (1.01 - 1.01i)T - 11iT^{2} \)
17 \( 1 + 1.39iT - 17T^{2} \)
19 \( 1 + (-3.94 - 3.94i)T + 19iT^{2} \)
23 \( 1 - 8.74T + 23T^{2} \)
29 \( 1 - 4.22T + 29T^{2} \)
31 \( 1 + (3.88 + 3.88i)T + 31iT^{2} \)
37 \( 1 + (-0.366 - 0.366i)T + 37iT^{2} \)
41 \( 1 + (-4.09 + 4.09i)T - 41iT^{2} \)
43 \( 1 - 9.18T + 43T^{2} \)
47 \( 1 + (-2.80 + 2.80i)T - 47iT^{2} \)
53 \( 1 + 5.94T + 53T^{2} \)
59 \( 1 + (6.02 - 6.02i)T - 59iT^{2} \)
61 \( 1 + 7.22T + 61T^{2} \)
67 \( 1 + (-1.78 - 1.78i)T + 67iT^{2} \)
71 \( 1 + (-7.68 - 7.68i)T + 71iT^{2} \)
73 \( 1 + (5.05 + 5.05i)T + 73iT^{2} \)
79 \( 1 + 8.51iT - 79T^{2} \)
83 \( 1 + (-6.91 - 6.91i)T + 83iT^{2} \)
89 \( 1 + (-4.69 - 4.69i)T + 89iT^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−10.35639315588869008129370115642, −9.436008829309064021951207322191, −9.061525279116298133951377895455, −7.72159485872900597663447752252, −7.17307638332051804698671832244, −5.80126312790125655897678678853, −4.74068187874633966849076054012, −3.45213854719311942020267458215, −2.45433860427207023302917449612, −1.14941420640615903126561690147, 0.920408144098068641257721072856, 3.18333943541176714490164332618, 4.38008460288239871478939518866, 5.14519189444823223334362925635, 6.39233096199959420062180269733, 7.06227479403596919046679555479, 7.83990219831261967350570656436, 9.092269200085324825374764618394, 9.526695656743601065193269398800, 10.53420205419039774261400391165

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