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Properties

Label 2-2e8-32.5-c3-0-20
Degree $2$
Conductor $256$
Sign $-0.822 - 0.568i$
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  + (−1.20 − 2.90i)3-s + (3.98 + 1.65i)5-s + (−22.4 − 22.4i)7-s + (12.1 − 12.1i)9-s + (−16.5 + 39.8i)11-s + (−17.9 + 7.42i)13-s − 13.5i·15-s + 45.9i·17-s + (−25.0 + 10.3i)19-s + (−38.0 + 91.9i)21-s + (40.3 − 40.3i)23-s + (−75.2 − 75.2i)25-s + (−128. − 53.0i)27-s + (88.6 + 214. i)29-s − 260.·31-s + ⋯
L(s)  = 1  + (−0.231 − 0.558i)3-s + (0.356 + 0.147i)5-s + (−1.20 − 1.20i)7-s + (0.448 − 0.448i)9-s + (−0.452 + 1.09i)11-s + (−0.382 + 0.158i)13-s − 0.233i·15-s + 0.656i·17-s + (−0.301 + 0.125i)19-s + (−0.395 + 0.955i)21-s + (0.365 − 0.365i)23-s + (−0.601 − 0.601i)25-s + (−0.912 − 0.378i)27-s + (0.567 + 1.37i)29-s − 1.50·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ -0.822 - 0.568i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07400779758\)
\(L(\frac12)\) \(\approx\) \(0.07400779758\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.20 + 2.90i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-3.98 - 1.65i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (22.4 + 22.4i)T + 343iT^{2} \)
11 \( 1 + (16.5 - 39.8i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (17.9 - 7.42i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 - 45.9iT - 4.91e3T^{2} \)
19 \( 1 + (25.0 - 10.3i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-40.3 + 40.3i)T - 1.21e4iT^{2} \)
29 \( 1 + (-88.6 - 214. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + 260.T + 2.97e4T^{2} \)
37 \( 1 + (70.4 + 29.1i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (251. - 251. i)T - 6.89e4iT^{2} \)
43 \( 1 + (95.7 - 231. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 15.5iT - 1.03e5T^{2} \)
53 \( 1 + (171. - 414. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (-53.3 - 22.0i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (297. + 718. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (377. + 911. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (-359. - 359. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-605. + 605. i)T - 3.89e5iT^{2} \)
79 \( 1 + 380. iT - 4.93e5T^{2} \)
83 \( 1 + (-235. + 97.4i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (949. + 949. i)T + 7.04e5iT^{2} \)
97 \( 1 - 663.T + 9.12e5T^{2} \)
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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

−10.77837530052635664732207000055, −10.07402119599915100506554488109, −9.363956227910256339814261073568, −7.70427721369564696028807603527, −6.88624622361440344780460436967, −6.29941509576013452859992457992, −4.62900202139089022552302736778, −3.40262653415418828461251492285, −1.68134506775727100860972927819, −0.02832211734688761862995908645, 2.35045675223695704741346726427, 3.57201830595650588909681018611, 5.25016639725737684819688752480, 5.78689312574696344000915867801, 7.09994652203513847929675006677, 8.503469994560334377344733944052, 9.425855332924878824663487070219, 10.06714628519757828512465066531, 11.14543319577426225256267373693, 12.11986873430851743617476736071

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