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Properties

Label 2-2e8-16.13-c1-0-6
Degree $2$
Conductor $256$
Sign $0.793 + 0.608i$
Analytic cond. $2.04417$
Root an. cond. $1.42974$
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  + (0.517 + 0.517i)3-s + (1.73 − 1.73i)5-s − 3.86i·7-s − 2.46i·9-s + (−3.34 + 3.34i)11-s + (0.267 + 0.267i)13-s + 1.79·15-s + 3.46·17-s + (3.34 + 3.34i)19-s + (1.99 − 1.99i)21-s + 1.79i·23-s − 0.999i·25-s + (2.82 − 2.82i)27-s + (1.73 + 1.73i)29-s − 5.65·31-s + ⋯
L(s)  = 1  + (0.298 + 0.298i)3-s + (0.774 − 0.774i)5-s − 1.46i·7-s − 0.821i·9-s + (−1.00 + 1.00i)11-s + (0.0743 + 0.0743i)13-s + 0.462·15-s + 0.840·17-s + (0.767 + 0.767i)19-s + (0.436 − 0.436i)21-s + 0.373i·23-s − 0.199i·25-s + (0.544 − 0.544i)27-s + (0.321 + 0.321i)29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40868 - 0.478184i\)
\(L(\frac12)\) \(\approx\) \(1.40868 - 0.478184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.517 - 0.517i)T + 3iT^{2} \)
5 \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \)
7 \( 1 + 3.86iT - 7T^{2} \)
11 \( 1 + (3.34 - 3.34i)T - 11iT^{2} \)
13 \( 1 + (-0.267 - 0.267i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (-3.34 - 3.34i)T + 19iT^{2} \)
23 \( 1 - 1.79iT - 23T^{2} \)
29 \( 1 + (-1.73 - 1.73i)T + 29iT^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (3.73 - 3.73i)T - 37iT^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + (-1.55 + 1.55i)T - 43iT^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (7.73 - 7.73i)T - 53iT^{2} \)
59 \( 1 + (-5.13 + 5.13i)T - 59iT^{2} \)
61 \( 1 + (3.73 + 3.73i)T + 61iT^{2} \)
67 \( 1 + (4.38 + 4.38i)T + 67iT^{2} \)
71 \( 1 - 1.79iT - 71T^{2} \)
73 \( 1 + 2.53iT - 73T^{2} \)
79 \( 1 + 4.14T + 79T^{2} \)
83 \( 1 + (1.55 + 1.55i)T + 83iT^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 - 3.46T + 97T^{2} \)
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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

−12.14666291712774243133541201996, −10.64449489439589226409434718112, −9.902166847184677512258556713914, −9.331105924271227178128862327632, −7.945361694478314477908390103717, −7.10044952128334916272933941725, −5.66032067777447841054328425749, −4.58455444510317181107199737570, −3.38320210369840373717375912793, −1.35382921789853654129474640576, 2.28714520577151807716240044699, 2.98831690792390589773093002338, 5.34415393204930442804594885715, 5.81525931473009583932336576531, 7.23165968723150063871320958430, 8.277192088862390876473861430172, 9.131748148261040884007891858100, 10.32333857860287657444695417889, 11.03869003597832218097710808271, 12.17228818062529872918202849850

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