Properties

Label 2-136-136.101-c1-0-9
Degree $2$
Conductor $136$
Sign $0.670 - 0.741i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.09 + 0.894i)2-s + 1.88·3-s + (0.400 + 1.95i)4-s − 2.41·5-s + (2.06 + 1.68i)6-s − 2.57i·7-s + (−1.31 + 2.50i)8-s + 0.564·9-s + (−2.64 − 2.15i)10-s − 0.736·11-s + (0.756 + 3.69i)12-s − 4.07i·13-s + (2.29 − 2.81i)14-s − 4.55·15-s + (−3.67 + 1.56i)16-s + (3.99 + 1.02i)17-s + ⋯
L(s)  = 1  + (0.774 + 0.632i)2-s + 1.09·3-s + (0.200 + 0.979i)4-s − 1.07·5-s + (0.844 + 0.689i)6-s − 0.971i·7-s + (−0.464 + 0.885i)8-s + 0.188·9-s + (−0.836 − 0.682i)10-s − 0.221·11-s + (0.218 + 1.06i)12-s − 1.13i·13-s + (0.614 − 0.752i)14-s − 1.17·15-s + (−0.919 + 0.392i)16-s + (0.968 + 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.670 - 0.741i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.670 - 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63725 + 0.726666i\)
\(L(\frac12)\) \(\approx\) \(1.63725 + 0.726666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.894i)T \)
17 \( 1 + (-3.99 - 1.02i)T \)
good3 \( 1 - 1.88T + 3T^{2} \)
5 \( 1 + 2.41T + 5T^{2} \)
7 \( 1 + 2.57iT - 7T^{2} \)
11 \( 1 + 0.736T + 11T^{2} \)
13 \( 1 + 4.07iT - 13T^{2} \)
19 \( 1 + 0.853iT - 19T^{2} \)
23 \( 1 - 8.20iT - 23T^{2} \)
29 \( 1 - 5.86T + 29T^{2} \)
31 \( 1 - 3.51iT - 31T^{2} \)
37 \( 1 + 4.38T + 37T^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + 1.09iT - 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 - 4.66iT - 53T^{2} \)
59 \( 1 - 12.5iT - 59T^{2} \)
61 \( 1 - 8.16T + 61T^{2} \)
67 \( 1 + 10.5iT - 67T^{2} \)
71 \( 1 - 0.511iT - 71T^{2} \)
73 \( 1 - 13.3iT - 73T^{2} \)
79 \( 1 + 1.45iT - 79T^{2} \)
83 \( 1 - 4.35iT - 83T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−13.67080963465926241941032890372, −12.61614242147944457725058712956, −11.60005636487600525787787245443, −10.31005193158426428881403210930, −8.664436524364988235427299414834, −7.78736402041580649452835917125, −7.29979723330198269967493990461, −5.44243141608483339860734891371, −3.87255242277334385840362261062, −3.18438639728596498808018829283, 2.40443976251949198853723594244, 3.53037113445320946768690074860, 4.78988517901047572031799521909, 6.41174707648920794650941228677, 7.974050639428045852476235947516, 8.889666669288098501932456488753, 9.993814996767281962003860679602, 11.48965035561529620843933862503, 12.01429293318177120736943794642, 13.03207994726407152519957797920

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