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Properties

Label 2-136-17.15-c3-0-2
Degree $2$
Conductor $136$
Sign $-0.913 - 0.407i$
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  + (3.26 + 7.87i)3-s + (−8.68 + 3.59i)5-s + (3.82 + 1.58i)7-s + (−32.3 + 32.3i)9-s + (−0.766 + 1.85i)11-s + 58.7i·13-s + (−56.7 − 56.7i)15-s + (−29.9 − 63.3i)17-s + (−59.0 − 59.0i)19-s + 35.2i·21-s + (−15.0 + 36.2i)23-s + (−25.8 + 25.8i)25-s + (−147. − 61.1i)27-s + (145. − 60.4i)29-s + (46.0 + 111. i)31-s + ⋯
L(s)  = 1  + (0.628 + 1.51i)3-s + (−0.777 + 0.321i)5-s + (0.206 + 0.0855i)7-s + (−1.19 + 1.19i)9-s + (−0.0210 + 0.0507i)11-s + 1.25i·13-s + (−0.976 − 0.976i)15-s + (−0.427 − 0.904i)17-s + (−0.712 − 0.712i)19-s + 0.366i·21-s + (−0.136 + 0.329i)23-s + (−0.206 + 0.206i)25-s + (−1.05 − 0.435i)27-s + (0.934 − 0.386i)29-s + (0.266 + 0.643i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.295511 + 1.38735i\)
\(L(\frac12)\) \(\approx\) \(0.295511 + 1.38735i\)
\(L(\frac{5}{2})\) 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 \( 1 + (29.9 + 63.3i)T \)
good3 \( 1 + (-3.26 - 7.87i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (8.68 - 3.59i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-3.82 - 1.58i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (0.766 - 1.85i)T + (-941. - 941. i)T^{2} \)
13 \( 1 - 58.7iT - 2.19e3T^{2} \)
19 \( 1 + (59.0 + 59.0i)T + 6.85e3iT^{2} \)
23 \( 1 + (15.0 - 36.2i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-145. + 60.4i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (-46.0 - 111. i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-129. - 312. i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-251. - 104. i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (169. - 169. i)T - 7.95e4iT^{2} \)
47 \( 1 - 150. iT - 1.03e5T^{2} \)
53 \( 1 + (-513. - 513. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-448. + 448. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-570. - 236. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 712.T + 3.00e5T^{2} \)
71 \( 1 + (317. + 767. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (471. - 195. i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-114. + 276. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (434. + 434. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.26e3iT - 7.04e5T^{2} \)
97 \( 1 + (949. - 393. i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−13.49422911048350686866362004181, −11.73079596486708792180761228411, −11.15699678286673566075917315710, −9.973419353813731543024883448846, −9.113571817396107189038443678301, −8.185542893088226274401437822714, −6.76924898903674223265353664726, −4.82766308002810200890720022481, −4.08627606781493887142545045052, −2.74772286507374031014954201841, 0.66275591091099976055946651244, 2.30345772826378292028453398866, 3.92869883159259643257289524300, 5.86832550367782564284911395474, 7.10379274189565165390859688796, 8.186283061269735516287268637117, 8.448796495828055524818462937597, 10.32029956276619069161231342688, 11.60590841908722389522372386687, 12.66172926560874339254450871621

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