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Properties

Label 2-136-8.5-c3-0-19
Degree $2$
Conductor $136$
Sign $0.893 + 0.448i$
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  + (−2.79 − 0.436i)2-s − 3.77i·3-s + (7.61 + 2.44i)4-s + 1.90i·5-s + (−1.64 + 10.5i)6-s + 11.0·7-s + (−20.2 − 10.1i)8-s + 12.7·9-s + (0.829 − 5.31i)10-s + 43.4i·11-s + (9.20 − 28.7i)12-s + 36.5i·13-s + (−30.7 − 4.80i)14-s + 7.16·15-s + (52.0 + 37.1i)16-s + 17·17-s + ⋯
L(s)  = 1  + (−0.988 − 0.154i)2-s − 0.725i·3-s + (0.952 + 0.305i)4-s + 0.170i·5-s + (−0.112 + 0.717i)6-s + 0.594·7-s + (−0.893 − 0.448i)8-s + 0.473·9-s + (0.0262 − 0.167i)10-s + 1.19i·11-s + (0.221 − 0.691i)12-s + 0.778i·13-s + (−0.587 − 0.0917i)14-s + 0.123·15-s + (0.813 + 0.580i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.893 + 0.448i$
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: \((2,\ 136,\ (\ :3/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.17893 - 0.279095i\)
\(L(\frac12)\) \(\approx\) \(1.17893 - 0.279095i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.79 + 0.436i)T \)
17 \( 1 - 17T \)
good3 \( 1 + 3.77iT - 27T^{2} \)
5 \( 1 - 1.90iT - 125T^{2} \)
7 \( 1 - 11.0T + 343T^{2} \)
11 \( 1 - 43.4iT - 1.33e3T^{2} \)
13 \( 1 - 36.5iT - 2.19e3T^{2} \)
19 \( 1 + 146. iT - 6.85e3T^{2} \)
23 \( 1 - 101.T + 1.21e4T^{2} \)
29 \( 1 - 95.5iT - 2.43e4T^{2} \)
31 \( 1 - 322.T + 2.97e4T^{2} \)
37 \( 1 + 125. iT - 5.06e4T^{2} \)
41 \( 1 + 253.T + 6.89e4T^{2} \)
43 \( 1 + 99.6iT - 7.95e4T^{2} \)
47 \( 1 - 125.T + 1.03e5T^{2} \)
53 \( 1 + 448. iT - 1.48e5T^{2} \)
59 \( 1 - 701. iT - 2.05e5T^{2} \)
61 \( 1 - 854. iT - 2.26e5T^{2} \)
67 \( 1 + 390. iT - 3.00e5T^{2} \)
71 \( 1 - 337.T + 3.57e5T^{2} \)
73 \( 1 - 96.1T + 3.89e5T^{2} \)
79 \( 1 - 131.T + 4.93e5T^{2} \)
83 \( 1 - 918. iT - 5.71e5T^{2} \)
89 \( 1 + 941.T + 7.04e5T^{2} \)
97 \( 1 + 60.3T + 9.12e5T^{2} \)
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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

−12.43036134898116977934583868675, −11.61843567649288471245606567665, −10.57429685215945500188475246099, −9.489667312776725853225633401603, −8.445472131942512510840621940611, −7.10329336654999240130437123235, −6.85583406803304050169771512447, −4.69371947234245965268792080937, −2.49172525920245014159014894833, −1.19131934998001875330621521423, 1.13875790542271089519760523704, 3.26072020086048159486037662731, 5.05515100949664014770346556272, 6.30807873838690652360002684593, 7.87509981926026292577128445510, 8.542779229590998183850442183705, 9.821783814699358994618008819225, 10.51944151657373689985417217003, 11.42025692275105973234845765231, 12.59976833500979222330179411397

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