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Properties

Label 2-136-17.2-c3-0-7
Degree $2$
Conductor $136$
Sign $-0.215 + 0.976i$
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  + (−6.48 + 2.68i)3-s + (1.11 + 2.69i)5-s + (−5.55 + 13.4i)7-s + (15.7 − 15.7i)9-s + (−10.4 − 4.33i)11-s − 48.9i·13-s + (−14.4 − 14.4i)15-s + (11.8 − 69.0i)17-s + (−50.5 − 50.5i)19-s − 101. i·21-s + (−35.0 − 14.5i)23-s + (82.3 − 82.3i)25-s + (12.6 − 30.4i)27-s + (−29.4 − 71.0i)29-s + (−60.2 + 24.9i)31-s + ⋯
L(s)  = 1  + (−1.24 + 0.517i)3-s + (0.0997 + 0.240i)5-s + (−0.299 + 0.723i)7-s + (0.583 − 0.583i)9-s + (−0.287 − 0.118i)11-s − 1.04i·13-s + (−0.249 − 0.249i)15-s + (0.169 − 0.985i)17-s + (−0.610 − 0.610i)19-s − 1.05i·21-s + (−0.317 − 0.131i)23-s + (0.659 − 0.659i)25-s + (0.0900 − 0.217i)27-s + (−0.188 − 0.455i)29-s + (−0.349 + 0.144i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.220044 - 0.273812i\)
\(L(\frac12)\) \(\approx\) \(0.220044 - 0.273812i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.8 + 69.0i)T \)
good3 \( 1 + (6.48 - 2.68i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-1.11 - 2.69i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (5.55 - 13.4i)T + (-242. - 242. i)T^{2} \)
11 \( 1 + (10.4 + 4.33i)T + (941. + 941. i)T^{2} \)
13 \( 1 + 48.9iT - 2.19e3T^{2} \)
19 \( 1 + (50.5 + 50.5i)T + 6.85e3iT^{2} \)
23 \( 1 + (35.0 + 14.5i)T + (8.60e3 + 8.60e3i)T^{2} \)
29 \( 1 + (29.4 + 71.0i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + (60.2 - 24.9i)T + (2.10e4 - 2.10e4i)T^{2} \)
37 \( 1 + (72.6 - 30.1i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (137. - 331. i)T + (-4.87e4 - 4.87e4i)T^{2} \)
43 \( 1 + (58.6 - 58.6i)T - 7.95e4iT^{2} \)
47 \( 1 - 11.8iT - 1.03e5T^{2} \)
53 \( 1 + (399. + 399. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-236. + 236. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-131. + 318. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + 634.T + 3.00e5T^{2} \)
71 \( 1 + (53.7 - 22.2i)T + (2.53e5 - 2.53e5i)T^{2} \)
73 \( 1 + (-87.9 - 212. i)T + (-2.75e5 + 2.75e5i)T^{2} \)
79 \( 1 + (-302. - 125. i)T + (3.48e5 + 3.48e5i)T^{2} \)
83 \( 1 + (1.01e3 + 1.01e3i)T + 5.71e5iT^{2} \)
89 \( 1 - 144. iT - 7.04e5T^{2} \)
97 \( 1 + (163. + 393. i)T + (-6.45e5 + 6.45e5i)T^{2} \)
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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

−12.27685100395456351081932849100, −11.36427207130287165272086481781, −10.52132781818529963683259776646, −9.622579605439192406498044567546, −8.240052409088894572667877647162, −6.66047582989870286887323025929, −5.67433242095688958983408601714, −4.77068486222913516400694852255, −2.85525361866168164904503712715, −0.20970724729317449819019716352, 1.50649046131752764121256568623, 3.98551857788446773534954765101, 5.40943386587506557912333205024, 6.46782770921478895902941085594, 7.35108737334466687950819932985, 8.851894330462467111071108522933, 10.27323974471355988712878215345, 11.00608610546951774996501310443, 12.12521859858110069488285698810, 12.78764974082203010417457301884

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