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Properties

Label 2-136-136.27-c3-0-5
Degree $2$
Conductor $136$
Sign $-0.238 - 0.971i$
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  + (−1.08 + 2.61i)2-s + (−3.92 − 2.62i)3-s + (−5.65 − 5.65i)4-s + (11.1 − 7.42i)6-s + (20.9 − 8.65i)8-s + (−1.78 − 4.31i)9-s + (13.5 + 20.2i)11-s + (7.37 + 37.0i)12-s + 64i·16-s + (32.4 + 62.1i)17-s + 13.2·18-s + (−63.1 + 152. i)19-s + (−67.4 + 13.4i)22-s + (−104. − 20.8i)24-s + (115. − 47.8i)25-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.756 − 0.505i)3-s + (−0.707 − 0.707i)4-s + (0.756 − 0.505i)6-s + (0.923 − 0.382i)8-s + (−0.0662 − 0.159i)9-s + (0.370 + 0.554i)11-s + (0.177 + 0.891i)12-s + i·16-s + (0.462 + 0.886i)17-s + 0.173·18-s + (−0.762 + 1.84i)19-s + (−0.653 + 0.130i)22-s + (−0.891 − 0.177i)24-s + (0.923 − 0.382i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.238 - 0.971i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.238 - 0.971i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.462684 + 0.589860i\)
\(L(\frac12)\) \(\approx\) \(0.462684 + 0.589860i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 2.61i)T \)
17 \( 1 + (-32.4 - 62.1i)T \)
good3 \( 1 + (3.92 + 2.62i)T + (10.3 + 24.9i)T^{2} \)
5 \( 1 + (-115. + 47.8i)T^{2} \)
7 \( 1 + (-316. - 131. i)T^{2} \)
11 \( 1 + (-13.5 - 20.2i)T + (-509. + 1.22e3i)T^{2} \)
13 \( 1 + 2.19e3iT^{2} \)
19 \( 1 + (63.1 - 152. i)T + (-4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-4.65e3 + 1.12e4i)T^{2} \)
29 \( 1 + (2.25e4 - 9.33e3i)T^{2} \)
31 \( 1 + (1.14e4 + 2.75e4i)T^{2} \)
37 \( 1 + (-1.93e4 - 4.67e4i)T^{2} \)
41 \( 1 + (-330. - 65.7i)T + (6.36e4 + 2.63e4i)T^{2} \)
43 \( 1 + (-128. - 310. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 1.03e5iT^{2} \)
53 \( 1 + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-21.4 + 8.89i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (2.09e5 + 8.68e4i)T^{2} \)
67 \( 1 - 354. iT - 3.00e5T^{2} \)
71 \( 1 + (-1.36e5 - 3.30e5i)T^{2} \)
73 \( 1 + (287. - 57.1i)T + (3.59e5 - 1.48e5i)T^{2} \)
79 \( 1 + (1.88e5 - 4.55e5i)T^{2} \)
83 \( 1 + (911. + 377. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (1.14e3 + 1.14e3i)T + 7.04e5iT^{2} \)
97 \( 1 + (-363. - 1.82e3i)T + (-8.43e5 + 3.49e5i)T^{2} \)
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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

−12.86450902572881855828941804839, −12.26134168105408926266702277261, −10.84194782430352997808273040042, −9.873452830909630893138582163908, −8.636788812492966434696300816466, −7.55087277436979824953090071095, −6.40633624966962300570097307015, −5.73912790285341986019356152633, −4.16975323905592879230160239645, −1.28845022484996971770214631377, 0.54546268475956683331415364452, 2.70646245477594528243838165596, 4.33331202813630615018002361015, 5.39898224645511219371948144703, 7.11395208338157315203552508648, 8.597901940965135270421724474322, 9.478985507228007754157822226950, 10.72603545544195839659307453956, 11.17477837562410012721720533555, 12.12568168246287910792769776038

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