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Properties

Label 2-136-17.13-c3-0-6
Degree $2$
Conductor $136$
Sign $0.704 + 0.709i$
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  + (−1.84 − 1.84i)3-s + (4.90 + 4.90i)5-s + (6.14 − 6.14i)7-s − 20.2i·9-s + (−19.4 + 19.4i)11-s + 64.7·13-s − 18.0i·15-s + (69.5 + 8.79i)17-s − 162. i·19-s − 22.6·21-s + (114. − 114. i)23-s − 76.9i·25-s + (−86.9 + 86.9i)27-s + (21.2 + 21.2i)29-s + (−11.9 − 11.9i)31-s + ⋯
L(s)  = 1  + (−0.354 − 0.354i)3-s + (0.438 + 0.438i)5-s + (0.331 − 0.331i)7-s − 0.748i·9-s + (−0.533 + 0.533i)11-s + 1.38·13-s − 0.310i·15-s + (0.992 + 0.125i)17-s − 1.96i·19-s − 0.235·21-s + (1.04 − 1.04i)23-s − 0.615i·25-s + (−0.619 + 0.619i)27-s + (0.135 + 0.135i)29-s + (−0.0690 − 0.0690i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.51312 - 0.629690i\)
\(L(\frac12)\) \(\approx\) \(1.51312 - 0.629690i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-69.5 - 8.79i)T \)
good3 \( 1 + (1.84 + 1.84i)T + 27iT^{2} \)
5 \( 1 + (-4.90 - 4.90i)T + 125iT^{2} \)
7 \( 1 + (-6.14 + 6.14i)T - 343iT^{2} \)
11 \( 1 + (19.4 - 19.4i)T - 1.33e3iT^{2} \)
13 \( 1 - 64.7T + 2.19e3T^{2} \)
19 \( 1 + 162. iT - 6.85e3T^{2} \)
23 \( 1 + (-114. + 114. i)T - 1.21e4iT^{2} \)
29 \( 1 + (-21.2 - 21.2i)T + 2.43e4iT^{2} \)
31 \( 1 + (11.9 + 11.9i)T + 2.97e4iT^{2} \)
37 \( 1 + (-206. - 206. i)T + 5.06e4iT^{2} \)
41 \( 1 + (164. - 164. i)T - 6.89e4iT^{2} \)
43 \( 1 - 150. iT - 7.95e4T^{2} \)
47 \( 1 + 371.T + 1.03e5T^{2} \)
53 \( 1 - 55.4iT - 1.48e5T^{2} \)
59 \( 1 - 15.6iT - 2.05e5T^{2} \)
61 \( 1 + (-127. + 127. i)T - 2.26e5iT^{2} \)
67 \( 1 - 159.T + 3.00e5T^{2} \)
71 \( 1 + (-27.7 - 27.7i)T + 3.57e5iT^{2} \)
73 \( 1 + (525. + 525. i)T + 3.89e5iT^{2} \)
79 \( 1 + (526. - 526. i)T - 4.93e5iT^{2} \)
83 \( 1 - 768. iT - 5.71e5T^{2} \)
89 \( 1 - 1.57e3T + 7.04e5T^{2} \)
97 \( 1 + (1.25e3 + 1.25e3i)T + 9.12e5iT^{2} \)
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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

−12.74607936229653777178417536142, −11.47940912082767532438663115477, −10.71776880603864489988306889473, −9.575813363020025592173367417598, −8.344481176101969221032777947320, −6.97101886198367428157664796721, −6.19615743092141628950777623994, −4.72222334335039377841328589789, −2.97419608193696650146584812316, −1.01589817565930965961255522807, 1.53154969192464018096912420351, 3.57853190685070924937633171417, 5.31525821455338546383278609289, 5.79843587888669929691028321379, 7.72986707438542947618846169745, 8.623977610248919127093411921399, 9.895566132447423151296634813369, 10.82902594532327399476514018558, 11.70780911590978783469719427629, 13.00255403623124410901975336109

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