Properties

Label 2-136-17.4-c3-0-4
Degree $2$
Conductor $136$
Sign $0.304 - 0.952i$
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  + (0.0399 − 0.0399i)3-s + (−1.97 + 1.97i)5-s + (−4.30 − 4.30i)7-s + 26.9i·9-s + (36.5 + 36.5i)11-s + 19.2·13-s + 0.158i·15-s + (−57.9 + 39.4i)17-s + 25.0i·19-s − 0.343·21-s + (105. + 105. i)23-s + 117. i·25-s + (2.15 + 2.15i)27-s + (−14.6 + 14.6i)29-s + (161. − 161. i)31-s + ⋯
L(s)  = 1  + (0.00769 − 0.00769i)3-s + (−0.177 + 0.177i)5-s + (−0.232 − 0.232i)7-s + 0.999i·9-s + (1.00 + 1.00i)11-s + 0.410·13-s + 0.00272i·15-s + (−0.826 + 0.563i)17-s + 0.301i·19-s − 0.00357·21-s + (0.959 + 0.959i)23-s + 0.937i·25-s + (0.0153 + 0.0153i)27-s + (−0.0940 + 0.0940i)29-s + (0.935 − 0.935i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.304 - 0.952i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.17155 + 0.855293i\)
\(L(\frac12)\) \(\approx\) \(1.17155 + 0.855293i\)
\(L(\frac{5}{2})\) 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 \( 1 + (57.9 - 39.4i)T \)
good3 \( 1 + (-0.0399 + 0.0399i)T - 27iT^{2} \)
5 \( 1 + (1.97 - 1.97i)T - 125iT^{2} \)
7 \( 1 + (4.30 + 4.30i)T + 343iT^{2} \)
11 \( 1 + (-36.5 - 36.5i)T + 1.33e3iT^{2} \)
13 \( 1 - 19.2T + 2.19e3T^{2} \)
19 \( 1 - 25.0iT - 6.85e3T^{2} \)
23 \( 1 + (-105. - 105. i)T + 1.21e4iT^{2} \)
29 \( 1 + (14.6 - 14.6i)T - 2.43e4iT^{2} \)
31 \( 1 + (-161. + 161. i)T - 2.97e4iT^{2} \)
37 \( 1 + (249. - 249. i)T - 5.06e4iT^{2} \)
41 \( 1 + (155. + 155. i)T + 6.89e4iT^{2} \)
43 \( 1 + 51.0iT - 7.95e4T^{2} \)
47 \( 1 - 261.T + 1.03e5T^{2} \)
53 \( 1 + 102. iT - 1.48e5T^{2} \)
59 \( 1 + 694. iT - 2.05e5T^{2} \)
61 \( 1 + (154. + 154. i)T + 2.26e5iT^{2} \)
67 \( 1 + 442.T + 3.00e5T^{2} \)
71 \( 1 + (-382. + 382. i)T - 3.57e5iT^{2} \)
73 \( 1 + (190. - 190. i)T - 3.89e5iT^{2} \)
79 \( 1 + (22.3 + 22.3i)T + 4.93e5iT^{2} \)
83 \( 1 - 669. iT - 5.71e5T^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + (-218. + 218. i)T - 9.12e5iT^{2} \)
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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

−13.08513422564920765331674126103, −11.84996943156947127090609586322, −10.93581081518437395667566717467, −9.898282950928659386812571311562, −8.768323735858833154097003307019, −7.50297822871348641591470519448, −6.55178737365624546354335251814, −4.97967712506638171825246227642, −3.68090103046209387060104221785, −1.78106861871324476516998552857, 0.77693773535520826600792917448, 3.06251982269827302078760352207, 4.39401589859771642932903003560, 6.08191947731073583829921637519, 6.88780254912224468916930261196, 8.705620989518892862628241647748, 9.057808530126462458685843325843, 10.57486226860323044301864974617, 11.65150103541620850398835864578, 12.39361635237024285996573544170

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