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Properties

Label 2-26e2-13.4-c3-0-33
Degree $2$
Conductor $676$
Sign $-0.964 - 0.265i$
Analytic cond. $39.8852$
Root an. cond. $6.31548$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.643 + 1.11i)3-s − 15.8i·5-s + (1.02 − 0.592i)7-s + (12.6 + 21.9i)9-s + (−33.6 − 19.4i)11-s + (17.7 + 10.2i)15-s + (−18.4 − 31.9i)17-s + (−37.1 + 21.4i)19-s + 1.52i·21-s + (101. − 176. i)23-s − 127.·25-s − 67.3·27-s + (−29.4 + 51.0i)29-s + 77.3i·31-s + (43.3 − 25.0i)33-s + ⋯
L(s)  = 1  + (−0.123 + 0.214i)3-s − 1.42i·5-s + (0.0553 − 0.0319i)7-s + (0.469 + 0.812i)9-s + (−0.923 − 0.533i)11-s + (0.304 + 0.176i)15-s + (−0.263 − 0.456i)17-s + (−0.449 + 0.259i)19-s + 0.0158i·21-s + (0.924 − 1.60i)23-s − 1.02·25-s − 0.480·27-s + (−0.188 + 0.327i)29-s + 0.448i·31-s + (0.228 − 0.132i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(39.8852\)
Root analytic conductor: \(6.31548\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :3/2),\ -0.964 - 0.265i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2987619638\)
\(L(\frac12)\) \(\approx\) \(0.2987619638\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.643 - 1.11i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 15.8iT - 125T^{2} \)
7 \( 1 + (-1.02 + 0.592i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (33.6 + 19.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (18.4 + 31.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (37.1 - 21.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-101. + 176. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (29.4 - 51.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 77.3iT - 2.97e4T^{2} \)
37 \( 1 + (223. + 129. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (146. + 84.8i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-183. - 317. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 249. iT - 1.03e5T^{2} \)
53 \( 1 - 157.T + 1.48e5T^{2} \)
59 \( 1 + (582. - 336. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (290. + 502. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (156. + 90.4i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-449. + 259. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 982. iT - 3.89e5T^{2} \)
79 \( 1 + 1.26e3T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + (1.28e3 + 742. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-303. + 175. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−9.558389458308507630127123936621, −8.666457026397308358227353530215, −8.125040529655148794780828907570, −7.07562202430471383429226764387, −5.74192513616067318300501503523, −4.90513841204860043332834275491, −4.39109588681330301172028632773, −2.75792967628977186145427521378, −1.38629942796279833672291346934, −0.084007303423178931087757874458, 1.77045518976126340279817302457, 2.95339217347809490428583889228, 3.87518529971314227295923782434, 5.24308025465482458962007586084, 6.32842345675673386914192669315, 7.06336394914500781203764529090, 7.63516641080405107965073628245, 8.911874778819976341508486300857, 9.926705235435929679306737035345, 10.51296379322594004011605607032

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