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Properties

Label 2-136-17.13-c3-0-1
Degree $2$
Conductor $136$
Sign $-0.0318 - 0.999i$
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  + (−0.657 − 0.657i)3-s + (13.8 + 13.8i)5-s + (−14.0 + 14.0i)7-s − 26.1i·9-s + (−29.5 + 29.5i)11-s − 19.0·13-s − 18.1i·15-s + (44.8 + 53.8i)17-s + 149. i·19-s + 18.4·21-s + (80.3 − 80.3i)23-s + 257. i·25-s + (−34.9 + 34.9i)27-s + (104. + 104. i)29-s + (−73.6 − 73.6i)31-s + ⋯
L(s)  = 1  + (−0.126 − 0.126i)3-s + (1.23 + 1.23i)5-s + (−0.756 + 0.756i)7-s − 0.968i·9-s + (−0.808 + 0.808i)11-s − 0.406·13-s − 0.312i·15-s + (0.640 + 0.768i)17-s + 1.80i·19-s + 0.191·21-s + (0.728 − 0.728i)23-s + 2.06i·25-s + (−0.248 + 0.248i)27-s + (0.666 + 0.666i)29-s + (−0.426 − 0.426i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0318 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.0318 - 0.999i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.0318 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.05225 + 1.08635i\)
\(L(\frac12)\) \(\approx\) \(1.05225 + 1.08635i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-44.8 - 53.8i)T \)
good3 \( 1 + (0.657 + 0.657i)T + 27iT^{2} \)
5 \( 1 + (-13.8 - 13.8i)T + 125iT^{2} \)
7 \( 1 + (14.0 - 14.0i)T - 343iT^{2} \)
11 \( 1 + (29.5 - 29.5i)T - 1.33e3iT^{2} \)
13 \( 1 + 19.0T + 2.19e3T^{2} \)
19 \( 1 - 149. iT - 6.85e3T^{2} \)
23 \( 1 + (-80.3 + 80.3i)T - 1.21e4iT^{2} \)
29 \( 1 + (-104. - 104. i)T + 2.43e4iT^{2} \)
31 \( 1 + (73.6 + 73.6i)T + 2.97e4iT^{2} \)
37 \( 1 + (196. + 196. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-283. + 283. i)T - 6.89e4iT^{2} \)
43 \( 1 + 20.5iT - 7.95e4T^{2} \)
47 \( 1 - 81.5T + 1.03e5T^{2} \)
53 \( 1 - 626. iT - 1.48e5T^{2} \)
59 \( 1 + 301. iT - 2.05e5T^{2} \)
61 \( 1 + (-352. + 352. i)T - 2.26e5iT^{2} \)
67 \( 1 - 924.T + 3.00e5T^{2} \)
71 \( 1 + (748. + 748. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-4.67 - 4.67i)T + 3.89e5iT^{2} \)
79 \( 1 + (-364. + 364. i)T - 4.93e5iT^{2} \)
83 \( 1 + 772. iT - 5.71e5T^{2} \)
89 \( 1 + 221.T + 7.04e5T^{2} \)
97 \( 1 + (7.27 + 7.27i)T + 9.12e5iT^{2} \)
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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

−12.67041534130793681444078885776, −12.39872109875935563504201701650, −10.61358835483639579856524257937, −10.02763966094827504245848268535, −9.120184286750800559501705362643, −7.38014360778552053837606667680, −6.29703741401231771359880060502, −5.63185693636609352541391603327, −3.34454353878830730612849607910, −2.13193294344358651017761447390, 0.76648040786663844881656930876, 2.70424213398463406586864733788, 4.84025243580622746357182178269, 5.49423158347284822149074776361, 7.00834498321218124899451542417, 8.388076998797200522059853844484, 9.515213038486700523937670336607, 10.20885156181745312376043822514, 11.37280246881728357804299547767, 12.96439572058654556781425451504

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