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Properties

Label 2-136-17.13-c3-0-3
Degree $2$
Conductor $136$
Sign $-0.498 - 0.866i$
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  + (5.56 + 5.56i)3-s + (−3.36 − 3.36i)5-s + (−19.4 + 19.4i)7-s + 34.9i·9-s + (−39.4 + 39.4i)11-s + 89.7·13-s − 37.4i·15-s + (−64.9 − 26.3i)17-s + 71.5i·19-s − 216.·21-s + (33.0 − 33.0i)23-s − 102. i·25-s + (−44.0 + 44.0i)27-s + (208. + 208. i)29-s + (46.5 + 46.5i)31-s + ⋯
L(s)  = 1  + (1.07 + 1.07i)3-s + (−0.300 − 0.300i)5-s + (−1.05 + 1.05i)7-s + 1.29i·9-s + (−1.08 + 1.08i)11-s + 1.91·13-s − 0.644i·15-s + (−0.926 − 0.376i)17-s + 0.863i·19-s − 2.25·21-s + (0.299 − 0.299i)23-s − 0.819i·25-s + (−0.313 + 0.313i)27-s + (1.33 + 1.33i)29-s + (0.269 + 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.498 - 0.866i$
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: \((2,\ 136,\ (\ :3/2),\ -0.498 - 0.866i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.880417 + 1.52206i\)
\(L(\frac12)\) \(\approx\) \(0.880417 + 1.52206i\)
\(L(\frac{5}{2})\) 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 \( 1 + (64.9 + 26.3i)T \)
good3 \( 1 + (-5.56 - 5.56i)T + 27iT^{2} \)
5 \( 1 + (3.36 + 3.36i)T + 125iT^{2} \)
7 \( 1 + (19.4 - 19.4i)T - 343iT^{2} \)
11 \( 1 + (39.4 - 39.4i)T - 1.33e3iT^{2} \)
13 \( 1 - 89.7T + 2.19e3T^{2} \)
19 \( 1 - 71.5iT - 6.85e3T^{2} \)
23 \( 1 + (-33.0 + 33.0i)T - 1.21e4iT^{2} \)
29 \( 1 + (-208. - 208. i)T + 2.43e4iT^{2} \)
31 \( 1 + (-46.5 - 46.5i)T + 2.97e4iT^{2} \)
37 \( 1 + (61.1 + 61.1i)T + 5.06e4iT^{2} \)
41 \( 1 + (131. - 131. i)T - 6.89e4iT^{2} \)
43 \( 1 + 57.3iT - 7.95e4T^{2} \)
47 \( 1 - 280.T + 1.03e5T^{2} \)
53 \( 1 + 239. iT - 1.48e5T^{2} \)
59 \( 1 + 1.44iT - 2.05e5T^{2} \)
61 \( 1 + (-411. + 411. i)T - 2.26e5iT^{2} \)
67 \( 1 - 618.T + 3.00e5T^{2} \)
71 \( 1 + (455. + 455. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-525. - 525. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-70.9 + 70.9i)T - 4.93e5iT^{2} \)
83 \( 1 - 399. iT - 5.71e5T^{2} \)
89 \( 1 + 616.T + 7.04e5T^{2} \)
97 \( 1 + (-877. - 877. i)T + 9.12e5iT^{2} \)
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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

−13.09378531899738442972456336154, −12.28021185886949414835475515434, −10.69582479971861734576393594370, −9.863333945153744701857842665254, −8.811139022170733980706058651178, −8.335973957998042844378637602109, −6.50591267393718894705780371056, −4.97261340039273355939217581548, −3.68303123834572965613592634693, −2.55539295186224484634760123410, 0.793452737883932713034680723369, 2.82134322448767674448824007081, 3.76217197091991599311837311936, 6.20051684804800063875000963293, 7.05554442280050451922900714037, 8.139176557672043747468703866466, 8.889090570532226678928842673434, 10.45105503771573179626935052797, 11.29808144763375254232882747624, 13.00987844775752889813152537393

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