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Properties

Label 2-136-136.75-c3-0-30
Degree $2$
Conductor $136$
Sign $-0.139 + 0.990i$
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 + (−5.14 + 7.69i)3-s + (−5.65 − 5.65i)4-s + (14.5 + 21.7i)6-s + (−20.9 + 8.65i)8-s + (−22.4 − 54.2i)9-s + (57.2 − 38.2i)11-s + (72.6 − 14.4i)12-s + 64i·16-s + (−32.4 − 62.1i)17-s − 165.·18-s + (63.1 − 152. i)19-s + (−37.9 − 190. i)22-s + (40.8 − 205. i)24-s + (−115. + 47.8i)25-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.989 + 1.48i)3-s + (−0.707 − 0.707i)4-s + (0.989 + 1.48i)6-s + (−0.923 + 0.382i)8-s + (−0.831 − 2.00i)9-s + (1.56 − 1.04i)11-s + (1.74 − 0.347i)12-s + i·16-s + (−0.462 − 0.886i)17-s − 2.17·18-s + (0.762 − 1.84i)19-s + (−0.367 − 1.84i)22-s + (0.347 − 1.74i)24-s + (−0.923 + 0.382i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.139 + 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.683136 - 0.786055i\)
\(L(\frac12)\) \(\approx\) \(0.683136 - 0.786055i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.14 - 7.69i)T + (-10.3 - 24.9i)T^{2} \)
5 \( 1 + (115. - 47.8i)T^{2} \)
7 \( 1 + (316. + 131. i)T^{2} \)
11 \( 1 + (-57.2 + 38.2i)T + (509. - 1.22e3i)T^{2} \)
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 \( 1 + (-78.5 + 394. i)T + (-6.36e4 - 2.63e4i)T^{2} \)
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 \( 1 + (236. + 1.18e3i)T + (-3.59e5 + 1.48e5i)T^{2} \)
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 \( 1 + (414. - 82.4i)T + (8.43e5 - 3.49e5i)T^{2} \)
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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

−11.83210802134313600809035832056, −11.45962186901894779412168366591, −10.73292603979127529507261549529, −9.333891749564327573740927921505, −9.184790100860610991665195406794, −6.44891889179004582804823899570, −5.34386484977851186144251965184, −4.34653870923956831555817930934, −3.28765422127457147600573518406, −0.56766301551703176094450394565, 1.54479769273759013644577738233, 4.14917825917106459293187340301, 5.73169398794425497615456215253, 6.44802996072948594853744307506, 7.33488983558121008560257808952, 8.270850619497227818501961087360, 9.809943199311410189453528500367, 11.58914352449361199502852137345, 12.22901810524094427609697257819, 12.88705303818538245433572108734

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