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Properties

Label 2-136-17.4-c3-0-5
Degree $2$
Conductor $136$
Sign $0.730 - 0.682i$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
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  + (3.07 − 3.07i)3-s + (−8.67 + 8.67i)5-s + (9.02 + 9.02i)7-s + 8.10i·9-s + (39.0 + 39.0i)11-s − 0.868·13-s + 53.3i·15-s + (69.8 − 6.21i)17-s − 50.0i·19-s + 55.4·21-s + (−34.6 − 34.6i)23-s − 25.5i·25-s + (107. + 107. i)27-s + (100. − 100. i)29-s + (−208. + 208. i)31-s + ⋯
L(s)  = 1  + (0.591 − 0.591i)3-s + (−0.776 + 0.776i)5-s + (0.487 + 0.487i)7-s + 0.300i·9-s + (1.07 + 1.07i)11-s − 0.0185·13-s + 0.918i·15-s + (0.996 − 0.0886i)17-s − 0.604i·19-s + 0.576·21-s + (−0.314 − 0.314i)23-s − 0.204i·25-s + (0.769 + 0.769i)27-s + (0.646 − 0.646i)29-s + (−1.20 + 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.730 - 0.682i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.730 - 0.682i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.71594 + 0.677142i\)
\(L(\frac12)\) \(\approx\) \(1.71594 + 0.677142i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-69.8 + 6.21i)T \)
good3 \( 1 + (-3.07 + 3.07i)T - 27iT^{2} \)
5 \( 1 + (8.67 - 8.67i)T - 125iT^{2} \)
7 \( 1 + (-9.02 - 9.02i)T + 343iT^{2} \)
11 \( 1 + (-39.0 - 39.0i)T + 1.33e3iT^{2} \)
13 \( 1 + 0.868T + 2.19e3T^{2} \)
19 \( 1 + 50.0iT - 6.85e3T^{2} \)
23 \( 1 + (34.6 + 34.6i)T + 1.21e4iT^{2} \)
29 \( 1 + (-100. + 100. i)T - 2.43e4iT^{2} \)
31 \( 1 + (208. - 208. i)T - 2.97e4iT^{2} \)
37 \( 1 + (92.7 - 92.7i)T - 5.06e4iT^{2} \)
41 \( 1 + (-127. - 127. i)T + 6.89e4iT^{2} \)
43 \( 1 - 274. iT - 7.95e4T^{2} \)
47 \( 1 + 92.6T + 1.03e5T^{2} \)
53 \( 1 + 460. iT - 1.48e5T^{2} \)
59 \( 1 + 539. iT - 2.05e5T^{2} \)
61 \( 1 + (456. + 456. i)T + 2.26e5iT^{2} \)
67 \( 1 + 105.T + 3.00e5T^{2} \)
71 \( 1 + (57.5 - 57.5i)T - 3.57e5iT^{2} \)
73 \( 1 + (-636. + 636. i)T - 3.89e5iT^{2} \)
79 \( 1 + (-448. - 448. i)T + 4.93e5iT^{2} \)
83 \( 1 + 264. iT - 5.71e5T^{2} \)
89 \( 1 + 1.57e3T + 7.04e5T^{2} \)
97 \( 1 + (-224. + 224. i)T - 9.12e5iT^{2} \)
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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

−12.74084345185189214337618424005, −11.91494397381322874549083777111, −11.00652126886862278418649505794, −9.678564104246101013859899847843, −8.410649896744123704211083860101, −7.52243119397078216567555704041, −6.68448308367197707429344874866, −4.84145170379827164323317663770, −3.28208904269142924386181510213, −1.81649218036859036917095185151, 0.980197940736618991093132327547, 3.54525720427025823482191912807, 4.23006694347591140278028572031, 5.83994431001579910619267882701, 7.54037337900416816227959936205, 8.539007248358387702343813932912, 9.259563229737069923026684881050, 10.53699252661742176026234429610, 11.71820052465528411440232741011, 12.42497145826445064479125684601

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