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Properties

Label 2-136-8.5-c3-0-31
Degree $2$
Conductor $136$
Sign $0.519 + 0.854i$
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  + (2.15 − 1.83i)2-s − 0.0638i·3-s + (1.26 − 7.89i)4-s + 9.14i·5-s + (−0.117 − 0.137i)6-s + 22.9·7-s + (−11.7 − 19.3i)8-s + 26.9·9-s + (16.7 + 19.6i)10-s + 0.742i·11-s + (−0.504 − 0.0810i)12-s − 74.3i·13-s + (49.4 − 42.1i)14-s + 0.583·15-s + (−60.7 − 20.0i)16-s + 17·17-s + ⋯
L(s)  = 1  + (0.761 − 0.648i)2-s − 0.0122i·3-s + (0.158 − 0.987i)4-s + 0.818i·5-s + (−0.00796 − 0.00934i)6-s + 1.24·7-s + (−0.519 − 0.854i)8-s + 0.999·9-s + (0.530 + 0.622i)10-s + 0.0203i·11-s + (−0.0121 − 0.00194i)12-s − 1.58i·13-s + (0.944 − 0.804i)14-s + 0.0100·15-s + (−0.949 − 0.313i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.45053 - 1.37791i\)
\(L(\frac12)\) \(\approx\) \(2.45053 - 1.37791i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.15 + 1.83i)T \)
17 \( 1 - 17T \)
good3 \( 1 + 0.0638iT - 27T^{2} \)
5 \( 1 - 9.14iT - 125T^{2} \)
7 \( 1 - 22.9T + 343T^{2} \)
11 \( 1 - 0.742iT - 1.33e3T^{2} \)
13 \( 1 + 74.3iT - 2.19e3T^{2} \)
19 \( 1 - 25.7iT - 6.85e3T^{2} \)
23 \( 1 + 148.T + 1.21e4T^{2} \)
29 \( 1 - 201. iT - 2.43e4T^{2} \)
31 \( 1 - 157.T + 2.97e4T^{2} \)
37 \( 1 - 318. iT - 5.06e4T^{2} \)
41 \( 1 - 138.T + 6.89e4T^{2} \)
43 \( 1 + 410. iT - 7.95e4T^{2} \)
47 \( 1 + 532.T + 1.03e5T^{2} \)
53 \( 1 - 109. iT - 1.48e5T^{2} \)
59 \( 1 - 14.4iT - 2.05e5T^{2} \)
61 \( 1 - 464. iT - 2.26e5T^{2} \)
67 \( 1 - 608. iT - 3.00e5T^{2} \)
71 \( 1 + 701.T + 3.57e5T^{2} \)
73 \( 1 + 862.T + 3.89e5T^{2} \)
79 \( 1 + 494.T + 4.93e5T^{2} \)
83 \( 1 - 121. iT - 5.71e5T^{2} \)
89 \( 1 + 545.T + 7.04e5T^{2} \)
97 \( 1 - 376.T + 9.12e5T^{2} \)
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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

−12.53523394544964699276842110416, −11.61525611313773579183589097135, −10.48690973521743006356220028842, −10.13420539859101420194412505557, −8.220138288683360565254494070170, −7.03623608752328235172005337993, −5.63825595340623744573691988618, −4.47342134360679896269253527736, −3.04945226215660710478936565907, −1.45152111654676262706628209911, 1.82616702696965974430898457474, 4.26366419150711402652972429160, 4.74043058619127938512421214788, 6.24919040830529021145288715281, 7.53372531646862212410952765774, 8.402692149951213081233303932630, 9.578625636940936561814717195520, 11.30905507272789194578712024874, 12.03502375955859955919415449101, 13.02318072854297568742736352360

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