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Properties

Label 2-2e8-32.29-c1-0-3
Degree $2$
Conductor $256$
Sign $0.938 - 0.344i$
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  + (1.27 + 0.529i)3-s + (0.707 + 1.70i)5-s + (2.74 − 2.74i)7-s + (−0.766 − 0.766i)9-s + (0.135 − 0.0560i)11-s + (−1.18 + 2.85i)13-s + 2.55i·15-s + 6.44i·17-s + (0.805 − 1.94i)19-s + (4.97 − 2.05i)21-s + (−0.749 − 0.749i)23-s + (1.12 − 1.12i)25-s + (−2.16 − 5.22i)27-s + (−4.32 − 1.79i)29-s − 1.17·31-s + ⋯
L(s)  = 1  + (0.738 + 0.305i)3-s + (0.316 + 0.763i)5-s + (1.03 − 1.03i)7-s + (−0.255 − 0.255i)9-s + (0.0408 − 0.0169i)11-s + (−0.327 + 0.790i)13-s + 0.660i·15-s + 1.56i·17-s + (0.184 − 0.445i)19-s + (1.08 − 0.449i)21-s + (−0.156 − 0.156i)23-s + (0.224 − 0.224i)25-s + (−0.416 − 1.00i)27-s + (−0.802 − 0.332i)29-s − 0.210·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67182 + 0.297041i\)
\(L(\frac12)\) \(\approx\) \(1.67182 + 0.297041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 0.529i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.74 + 2.74i)T - 7iT^{2} \)
11 \( 1 + (-0.135 + 0.0560i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.18 - 2.85i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.44iT - 17T^{2} \)
19 \( 1 + (-0.805 + 1.94i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.749 + 0.749i)T + 23iT^{2} \)
29 \( 1 + (4.32 + 1.79i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + (1.73 + 4.18i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.49 + 2.49i)T + 41iT^{2} \)
43 \( 1 + (6.10 - 2.52i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.66iT - 47T^{2} \)
53 \( 1 + (1.64 - 0.682i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.43 - 3.47i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.46 - 1.43i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (14.0 + 5.83i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-3.40 + 3.40i)T - 71iT^{2} \)
73 \( 1 + (0.442 + 0.442i)T + 73iT^{2} \)
79 \( 1 + 7.07iT - 79T^{2} \)
83 \( 1 + (-2.99 + 7.23i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.21 - 4.21i)T - 89iT^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−11.89810985348530141159974276667, −10.92685711114758469994341651873, −10.27324887667947778649449829767, −9.161967179919410596450565703305, −8.194260886401418140263736456014, −7.23594950405168605209951257712, −6.13920885477016796969744594649, −4.50992329846976011838640936671, −3.52541632529578599828463967728, −1.98036305591451127709235678780, 1.77717380933825874944991177655, 2.99120739169023281054858749985, 5.02603819273900233487019414136, 5.47364135744340532531006203128, 7.31807333582494663485633768881, 8.254015238290389149144177175080, 8.853297409573661407230806243786, 9.784934043621118561176692821407, 11.23507032222501518400997672596, 11.99073123946016827355866764403

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