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Properties

Label 2-2e8-32.13-c3-0-21
Degree $2$
Conductor $256$
Sign $-0.935 + 0.353i$
Analytic cond. $15.1044$
Root an. cond. $3.88644$
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  + (3.54 − 8.56i)3-s + (7.55 − 3.12i)5-s + (−7.16 + 7.16i)7-s + (−41.6 − 41.6i)9-s + (0.758 + 1.83i)11-s + (−71.0 − 29.4i)13-s − 75.7i·15-s − 98.5i·17-s + (89.5 + 37.1i)19-s + (35.9 + 86.7i)21-s + (−24.9 − 24.9i)23-s + (−41.1 + 41.1i)25-s + (−273. + 113. i)27-s + (57.8 − 139. i)29-s − 58.0·31-s + ⋯
L(s)  = 1  + (0.682 − 1.64i)3-s + (0.675 − 0.279i)5-s + (−0.386 + 0.386i)7-s + (−1.54 − 1.54i)9-s + (0.0207 + 0.0501i)11-s + (−1.51 − 0.628i)13-s − 1.30i·15-s − 1.40i·17-s + (1.08 + 0.448i)19-s + (0.373 + 0.901i)21-s + (−0.226 − 0.226i)23-s + (−0.329 + 0.329i)25-s + (−1.95 + 0.808i)27-s + (0.370 − 0.894i)29-s − 0.336·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.935 + 0.353i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.886697728\)
\(L(\frac12)\) \(\approx\) \(1.886697728\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-3.54 + 8.56i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-7.55 + 3.12i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (7.16 - 7.16i)T - 343iT^{2} \)
11 \( 1 + (-0.758 - 1.83i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (71.0 + 29.4i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 98.5iT - 4.91e3T^{2} \)
19 \( 1 + (-89.5 - 37.1i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (24.9 + 24.9i)T + 1.21e4iT^{2} \)
29 \( 1 + (-57.8 + 139. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 58.0T + 2.97e4T^{2} \)
37 \( 1 + (-202. + 84.0i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (45.3 + 45.3i)T + 6.89e4iT^{2} \)
43 \( 1 + (-89.7 - 216. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 4.38iT - 1.03e5T^{2} \)
53 \( 1 + (8.98 + 21.6i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-287. + 119. i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-28.2 + 68.0i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-293. + 708. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-579. + 579. i)T - 3.57e5iT^{2} \)
73 \( 1 + (258. + 258. i)T + 3.89e5iT^{2} \)
79 \( 1 - 834. iT - 4.93e5T^{2} \)
83 \( 1 + (-234. - 97.3i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-179. + 179. i)T - 7.04e5iT^{2} \)
97 \( 1 + 624.T + 9.12e5T^{2} \)
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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

−11.60827022891507685479681106878, −9.742509045998857125660580006311, −9.296531552227087388142529397385, −7.906263331873942875289134937478, −7.36976305090457413551906702054, −6.23943861904705023245208165885, −5.22507835708545416596679979459, −2.97121278118978274589695969683, −2.16011847776186792722783217109, −0.64901050242731005313108868376, 2.35357817925284677868950590418, 3.53455335193044029398933644097, 4.57758970677929761087699402390, 5.64302208959935759659451497709, 7.09025478885541779275201688079, 8.410397102581162985659239220876, 9.475445552508583460288370467592, 9.941161779529350489353601110353, 10.60973618489816291385816489950, 11.78236319678593973011738795649

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