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Properties

Label 2-2e8-32.13-c1-0-3
Degree $2$
Conductor $256$
Sign $0.941 + 0.336i$
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  + (0.0794 − 0.191i)3-s + (−0.707 + 0.292i)5-s + (2.27 − 2.27i)7-s + (2.09 + 2.09i)9-s + (−1.49 − 3.60i)11-s + (4.50 + 1.86i)13-s + 0.158i·15-s − 3.05i·17-s + (3.87 + 1.60i)19-s + (−0.255 − 0.616i)21-s + (−0.271 − 0.271i)23-s + (−3.12 + 3.12i)25-s + (1.14 − 0.473i)27-s + (0.931 − 2.24i)29-s − 6.82·31-s + ⋯
L(s)  = 1  + (0.0458 − 0.110i)3-s + (−0.316 + 0.130i)5-s + (0.858 − 0.858i)7-s + (0.696 + 0.696i)9-s + (−0.450 − 1.08i)11-s + (1.24 + 0.517i)13-s + 0.0410i·15-s − 0.740i·17-s + (0.889 + 0.368i)19-s + (−0.0557 − 0.134i)21-s + (−0.0565 − 0.0565i)23-s + (−0.624 + 0.624i)25-s + (0.219 − 0.0911i)27-s + (0.173 − 0.417i)29-s − 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(2.04417\)
Root analytic conductor: \(1.42974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 256,\ (\ :1/2),\ 0.941 + 0.336i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35240 - 0.234464i\)
\(L(\frac12)\) \(\approx\) \(1.35240 - 0.234464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0794 + 0.191i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.707 - 0.292i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.27 + 2.27i)T - 7iT^{2} \)
11 \( 1 + (1.49 + 3.60i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-4.50 - 1.86i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 3.05iT - 17T^{2} \)
19 \( 1 + (-3.87 - 1.60i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.271 + 0.271i)T + 23iT^{2} \)
29 \( 1 + (-0.931 + 2.24i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 + (3.63 - 1.50i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.54 + 1.54i)T + 41iT^{2} \)
43 \( 1 + (-0.748 - 1.80i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.37iT - 47T^{2} \)
53 \( 1 + (1.67 + 4.04i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (10.1 - 4.19i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.35 - 3.28i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.99 - 4.81i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (6.47 - 6.47i)T - 71iT^{2} \)
73 \( 1 + (2.84 + 2.84i)T + 73iT^{2} \)
79 \( 1 + 9.74iT - 79T^{2} \)
83 \( 1 + (9.04 + 3.74i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-7.58 + 7.58i)T - 89iT^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
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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

−11.65195184393192998493270889072, −11.08233158534727393182255137470, −10.30577819758202572965738782161, −8.949877267704611786915119741662, −7.85235432410898189757078336111, −7.31422842630145979342370953436, −5.83012469178559654632473715545, −4.58965041661086261495962137298, −3.44392181827314499916270087398, −1.43127219942881954151784623146, 1.73629161549614519628228408764, 3.57436855186128655525494349831, 4.79882858504322103384065343789, 5.89427226984086239953643776920, 7.24197759067538263616117200934, 8.227125999236080816200832243645, 9.105041288624363610639653425994, 10.18262311406597823321737825143, 11.16151138735189316904870141562, 12.18094348659733907485196943062

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