Properties

Label 2-136-17.16-c3-0-6
Degree $2$
Conductor $136$
Sign $0.782 - 0.622i$
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  + 2.62i·3-s − 11.5i·5-s + 23.0i·7-s + 20.1·9-s + 1.19i·11-s + 58.3·13-s + 30.2·15-s + (−54.8 + 43.6i)17-s + 138.·19-s − 60.4·21-s + 180. i·23-s − 7.98·25-s + 123. i·27-s + 115. i·29-s − 210. i·31-s + ⋯
L(s)  = 1  + 0.504i·3-s − 1.03i·5-s + 1.24i·7-s + 0.745·9-s + 0.0327i·11-s + 1.24·13-s + 0.520·15-s + (−0.782 + 0.622i)17-s + 1.67·19-s − 0.627·21-s + 1.64i·23-s − 0.0638·25-s + 0.880i·27-s + 0.736i·29-s − 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68076 + 0.586758i\)
\(L(\frac12)\) \(\approx\) \(1.68076 + 0.586758i\)
\(L(\frac{5}{2})\) 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 \( 1 + (54.8 - 43.6i)T \)
good3 \( 1 - 2.62iT - 27T^{2} \)
5 \( 1 + 11.5iT - 125T^{2} \)
7 \( 1 - 23.0iT - 343T^{2} \)
11 \( 1 - 1.19iT - 1.33e3T^{2} \)
13 \( 1 - 58.3T + 2.19e3T^{2} \)
19 \( 1 - 138.T + 6.85e3T^{2} \)
23 \( 1 - 180. iT - 1.21e4T^{2} \)
29 \( 1 - 115. iT - 2.43e4T^{2} \)
31 \( 1 + 210. iT - 2.97e4T^{2} \)
37 \( 1 + 210. iT - 5.06e4T^{2} \)
41 \( 1 + 297. iT - 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + 199.T + 1.03e5T^{2} \)
53 \( 1 + 706.T + 1.48e5T^{2} \)
59 \( 1 + 182.T + 2.05e5T^{2} \)
61 \( 1 - 489. iT - 2.26e5T^{2} \)
67 \( 1 + 167.T + 3.00e5T^{2} \)
71 \( 1 + 818. iT - 3.57e5T^{2} \)
73 \( 1 + 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 + 110. iT - 4.93e5T^{2} \)
83 \( 1 - 118.T + 5.71e5T^{2} \)
89 \( 1 - 400.T + 7.04e5T^{2} \)
97 \( 1 - 924. iT - 9.12e5T^{2} \)
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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

−12.83809028253263988130793244945, −11.90412383666398273629872678022, −10.86431362976493171304382198687, −9.339955632610303486266926163705, −9.030054106137236379466249608401, −7.68294401288215629448384176458, −5.94148685814517768947610144992, −5.00895300638379963717581892643, −3.63545946294933430942593375005, −1.52255253395124227206334714495, 1.10222334703488268174480279219, 3.09807130355150348641933600340, 4.47876022020932137234349562861, 6.50475756465951423525505010232, 7.02608184319071590435899080800, 8.121738153984586735638281539542, 9.754773591310759185388130246400, 10.66869468150050434325391888166, 11.42114461196812305156892536426, 12.82988161074864512167161257058

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