Properties

Label 2-136-17.13-c3-0-4
Degree $2$
Conductor $136$
Sign $-0.181 - 0.983i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (6.66 + 6.66i)3-s + (3.32 + 3.32i)5-s + (−12.8 + 12.8i)7-s + 61.7i·9-s + (30.6 − 30.6i)11-s − 42.1·13-s + 44.2i·15-s + (52.4 + 46.5i)17-s − 112. i·19-s − 171.·21-s + (−49.9 + 49.9i)23-s − 102. i·25-s + (−231. + 231. i)27-s + (134. + 134. i)29-s + (75.2 + 75.2i)31-s + ⋯
L(s)  = 1  + (1.28 + 1.28i)3-s + (0.297 + 0.297i)5-s + (−0.693 + 0.693i)7-s + 2.28i·9-s + (0.841 − 0.841i)11-s − 0.900·13-s + 0.761i·15-s + (0.747 + 0.663i)17-s − 1.35i·19-s − 1.77·21-s + (−0.452 + 0.452i)23-s − 0.823i·25-s + (−1.65 + 1.65i)27-s + (0.862 + 0.862i)29-s + (0.435 + 0.435i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.181 - 0.983i$
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: \((2,\ 136,\ (\ :3/2),\ -0.181 - 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52585 + 1.83253i\)
\(L(\frac12)\) \(\approx\) \(1.52585 + 1.83253i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-52.4 - 46.5i)T \)
good3 \( 1 + (-6.66 - 6.66i)T + 27iT^{2} \)
5 \( 1 + (-3.32 - 3.32i)T + 125iT^{2} \)
7 \( 1 + (12.8 - 12.8i)T - 343iT^{2} \)
11 \( 1 + (-30.6 + 30.6i)T - 1.33e3iT^{2} \)
13 \( 1 + 42.1T + 2.19e3T^{2} \)
19 \( 1 + 112. iT - 6.85e3T^{2} \)
23 \( 1 + (49.9 - 49.9i)T - 1.21e4iT^{2} \)
29 \( 1 + (-134. - 134. i)T + 2.43e4iT^{2} \)
31 \( 1 + (-75.2 - 75.2i)T + 2.97e4iT^{2} \)
37 \( 1 + (-246. - 246. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-170. + 170. i)T - 6.89e4iT^{2} \)
43 \( 1 + 112. iT - 7.95e4T^{2} \)
47 \( 1 + 188.T + 1.03e5T^{2} \)
53 \( 1 + 642. iT - 1.48e5T^{2} \)
59 \( 1 + 431. iT - 2.05e5T^{2} \)
61 \( 1 + (41.0 - 41.0i)T - 2.26e5iT^{2} \)
67 \( 1 - 173.T + 3.00e5T^{2} \)
71 \( 1 + (232. + 232. i)T + 3.57e5iT^{2} \)
73 \( 1 + (197. + 197. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-936. + 936. i)T - 4.93e5iT^{2} \)
83 \( 1 + 168. iT - 5.71e5T^{2} \)
89 \( 1 - 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + (-1.20e3 - 1.20e3i)T + 9.12e5iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.34240010631776353231385749065, −11.98022739515787699012428739523, −10.60592765265596217541313731746, −9.721521264220116145669377380134, −9.058080030499228807570945185654, −8.118465540068987840661139754136, −6.42579701438807851002475874341, −4.91426045225014540049169974058, −3.48741780355681909089024958885, −2.58979619819620027243442167496, 1.15161978656404496336489005028, 2.58346040917408728412301194428, 4.03273805617324873154415317055, 6.21704494862749629114431227554, 7.27233480578363082469675963052, 7.948426030195185963452112487533, 9.400890172044165868103748506831, 9.868838128574571707947990532697, 12.01736527352078966944397400389, 12.56978540648662001257851211748

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