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Properties

Label 2-26e2-52.19-c1-0-5
Degree $2$
Conductor $676$
Sign $-0.546 - 0.837i$
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  + (−1.08 + 0.902i)2-s + (−0.736 − 0.425i)3-s + (0.372 − 1.96i)4-s + (0.166 − 0.166i)5-s + (1.18 − 0.201i)6-s + (−2.55 − 0.684i)7-s + (1.36 + 2.47i)8-s + (−1.13 − 1.97i)9-s + (−0.0311 + 0.331i)10-s + (0.373 + 1.39i)11-s + (−1.10 + 1.28i)12-s + (3.39 − 1.55i)14-s + (−0.193 + 0.0517i)15-s + (−3.72 − 1.46i)16-s + (−1.21 + 0.699i)17-s + (3.01 + 1.12i)18-s + ⋯
L(s)  = 1  + (−0.770 + 0.637i)2-s + (−0.425 − 0.245i)3-s + (0.186 − 0.982i)4-s + (0.0744 − 0.0744i)5-s + (0.483 − 0.0821i)6-s + (−0.965 − 0.258i)7-s + (0.483 + 0.875i)8-s + (−0.379 − 0.657i)9-s + (−0.00984 + 0.104i)10-s + (0.112 + 0.419i)11-s + (−0.320 + 0.371i)12-s + (0.908 − 0.416i)14-s + (−0.0498 + 0.0133i)15-s + (−0.930 − 0.366i)16-s + (−0.293 + 0.169i)17-s + (0.711 + 0.264i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.546 - 0.837i$
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: \((2,\ 676,\ (\ :1/2),\ -0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185091 + 0.341997i\)
\(L(\frac12)\) \(\approx\) \(0.185091 + 0.341997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.902i)T \)
13 \( 1 \)
good3 \( 1 + (0.736 + 0.425i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.166 + 0.166i)T - 5iT^{2} \)
7 \( 1 + (2.55 + 0.684i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.373 - 1.39i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.21 - 0.699i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.44 - 5.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.37 + 7.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.11 - 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.88 - 3.88i)T + 31iT^{2} \)
37 \( 1 + (0.5 - 0.133i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.5 - 5.59i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.59 - 7.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.80 - 2.80i)T - 47iT^{2} \)
53 \( 1 + 5.94T + 53T^{2} \)
59 \( 1 + (8.22 + 2.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.61 - 6.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.43 + 0.652i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.81 - 10.5i)T + (-61.4 - 35.5i)T^{2} \)
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 \( 1 + (6.41 - 1.71i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (14.9 + 4.00i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.61576554702420276137917772611, −9.783900562613397537321513688901, −9.094305980774659788061233189660, −8.238346118206672812592864122591, −7.07277575903713057112365816718, −6.49329912225675637805023029948, −5.81156073865779985359784324154, −4.57352283593984535141488436371, −3.05231744597326366232344944217, −1.27756335469795298859345911901, 0.30059448346164147504660401106, 2.30494191507820431096181209319, 3.24368473857560257443271703560, 4.51072848944490096931575243225, 5.77812898334347081060886450264, 6.74533169236185151030006673795, 7.71211834012972656986679407314, 8.762301567213008409358110428403, 9.409300218732227155751534876173, 10.20764884201092211227099809105

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