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Properties

Label 2-2e8-32.21-c3-0-3
Degree $2$
Conductor $256$
Sign $0.423 - 0.905i$
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  + (−9.18 + 3.80i)3-s + (−1.04 + 2.51i)5-s + (−16.1 − 16.1i)7-s + (50.8 − 50.8i)9-s + (3.72 + 1.54i)11-s + (−9.23 − 22.3i)13-s − 27.0i·15-s + 4.95i·17-s + (−26.0 − 62.8i)19-s + (209. + 86.6i)21-s + (−82.8 + 82.8i)23-s + (83.1 + 83.1i)25-s + (−170. + 412. i)27-s + (−150. + 62.3i)29-s + 141.·31-s + ⋯
L(s)  = 1  + (−1.76 + 0.732i)3-s + (−0.0932 + 0.225i)5-s + (−0.869 − 0.869i)7-s + (1.88 − 1.88i)9-s + (0.102 + 0.0422i)11-s + (−0.197 − 0.475i)13-s − 0.466i·15-s + 0.0706i·17-s + (−0.314 − 0.758i)19-s + (2.17 + 0.900i)21-s + (−0.750 + 0.750i)23-s + (0.665 + 0.665i)25-s + (−1.21 + 2.93i)27-s + (−0.963 + 0.399i)29-s + 0.820·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.423 - 0.905i$
Analytic conductor: \(15.1044\)
Root analytic conductor: \(3.88644\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :3/2),\ 0.423 - 0.905i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5799241854\)
\(L(\frac12)\) \(\approx\) \(0.5799241854\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.18 - 3.80i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (1.04 - 2.51i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (16.1 + 16.1i)T + 343iT^{2} \)
11 \( 1 + (-3.72 - 1.54i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (9.23 + 22.3i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 4.95iT - 4.91e3T^{2} \)
19 \( 1 + (26.0 + 62.8i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (82.8 - 82.8i)T - 1.21e4iT^{2} \)
29 \( 1 + (150. - 62.3i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 141.T + 2.97e4T^{2} \)
37 \( 1 + (-1.05 + 2.55i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-8.70 + 8.70i)T - 6.89e4iT^{2} \)
43 \( 1 + (-290. - 120. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 450. iT - 1.03e5T^{2} \)
53 \( 1 + (-114. - 47.4i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-124. + 300. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (-223. + 92.7i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-204. + 84.5i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-606. - 606. i)T + 3.57e5iT^{2} \)
73 \( 1 + (531. - 531. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.12e3iT - 4.93e5T^{2} \)
83 \( 1 + (118. + 286. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (-191. - 191. i)T + 7.04e5iT^{2} \)
97 \( 1 + 38.4T + 9.12e5T^{2} \)
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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

−11.48043260842512538874704330289, −10.83338754338271450551140198786, −10.07395767850607658107412664007, −9.347531355364603049637085298720, −7.38731933880088795955767569558, −6.57514520742589507949516054641, −5.65892406419002215205333429617, −4.53566479267436904218368702004, −3.51385530349083418395447905100, −0.76801667440550681695819093058, 0.45249290796378539502856336302, 2.13391462520345266346962984101, 4.31782336960521093621778768348, 5.56741645381427954625952744224, 6.23340271076239162815535386231, 7.05308773647662151400520559965, 8.362093251388562600549064009918, 9.739254795497274674011400548911, 10.61192361686811544056016129337, 11.69067084903871997793207934285

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