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Properties

Label 2-136-17.16-c3-0-12
Degree $2$
Conductor $136$
Sign $-0.869 + 0.494i$
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  − 4.49i·3-s − 18.9i·5-s + 7.78i·7-s + 6.80·9-s − 11.1i·11-s − 79.5·13-s − 85.3·15-s + (−60.9 + 34.6i)17-s + 104.·19-s + 35.0·21-s − 105. i·23-s − 235.·25-s − 151. i·27-s − 193. i·29-s + 258. i·31-s + ⋯
L(s)  = 1  − 0.864i·3-s − 1.69i·5-s + 0.420i·7-s + 0.251·9-s − 0.306i·11-s − 1.69·13-s − 1.46·15-s + (−0.869 + 0.494i)17-s + 1.26·19-s + 0.363·21-s − 0.960i·23-s − 1.88·25-s − 1.08i·27-s − 1.23i·29-s + 1.49i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.869 + 0.494i$
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: \((2,\ 136,\ (\ :3/2),\ -0.869 + 0.494i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.331784 - 1.25363i\)
\(L(\frac12)\) \(\approx\) \(0.331784 - 1.25363i\)
\(L(\frac{5}{2})\) 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 \( 1 + (60.9 - 34.6i)T \)
good3 \( 1 + 4.49iT - 27T^{2} \)
5 \( 1 + 18.9iT - 125T^{2} \)
7 \( 1 - 7.78iT - 343T^{2} \)
11 \( 1 + 11.1iT - 1.33e3T^{2} \)
13 \( 1 + 79.5T + 2.19e3T^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 + 105. iT - 1.21e4T^{2} \)
29 \( 1 + 193. iT - 2.43e4T^{2} \)
31 \( 1 - 258. iT - 2.97e4T^{2} \)
37 \( 1 + 35.4iT - 5.06e4T^{2} \)
41 \( 1 + 132. iT - 6.89e4T^{2} \)
43 \( 1 + 214.T + 7.95e4T^{2} \)
47 \( 1 - 339.T + 1.03e5T^{2} \)
53 \( 1 - 124.T + 1.48e5T^{2} \)
59 \( 1 - 735.T + 2.05e5T^{2} \)
61 \( 1 + 234. iT - 2.26e5T^{2} \)
67 \( 1 - 489.T + 3.00e5T^{2} \)
71 \( 1 + 265. iT - 3.57e5T^{2} \)
73 \( 1 - 193. iT - 3.89e5T^{2} \)
79 \( 1 + 852. iT - 4.93e5T^{2} \)
83 \( 1 + 33.8T + 5.71e5T^{2} \)
89 \( 1 - 13.3T + 7.04e5T^{2} \)
97 \( 1 + 861. iT - 9.12e5T^{2} \)
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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

−12.35061479388629109931077760280, −11.87774875041506270430597465060, −10.04236190770253143333925551466, −9.004234465589843228706187115909, −8.101104048292026954492483170848, −6.98776829576331261174680684743, −5.48298562755154300480753654903, −4.48286729642331337712773147010, −2.13850578307480626133200623243, −0.64611094431313316288015155097, 2.58783278209091631197643201291, 3.86147254893944536923568093289, 5.20967343480169653661630046056, 7.01062539727001893624857690481, 7.39667955082419083960178049776, 9.549480680157053231410016684317, 10.00603848895875762545744145384, 10.96556898228214904833162785060, 11.84014122729980454064430714160, 13.38599431310211325260213542907

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