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Properties

Label 2-136-17.2-c1-0-4
Degree $2$
Conductor $136$
Sign $0.651 + 0.758i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 0.414i)3-s + (−1.29 − 3.12i)5-s + (1 − 2.41i)7-s + (−1.29 + 1.29i)9-s + (4.41 + 1.82i)11-s + 5.41i·13-s + (−2.58 − 2.58i)15-s + (−2.82 − 3i)17-s + (2.58 + 2.58i)19-s − 2.82i·21-s + (−0.414 − 0.171i)23-s + (−4.53 + 4.53i)25-s + (−2 + 4.82i)27-s + (−0.878 − 2.12i)29-s + (2.41 − i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.239i)3-s + (−0.578 − 1.39i)5-s + (0.377 − 0.912i)7-s + (−0.430 + 0.430i)9-s + (1.33 + 0.551i)11-s + 1.50i·13-s + (−0.667 − 0.667i)15-s + (−0.685 − 0.727i)17-s + (0.593 + 0.593i)19-s − 0.617i·21-s + (−0.0863 − 0.0357i)23-s + (−0.907 + 0.907i)25-s + (−0.384 + 0.929i)27-s + (−0.163 − 0.393i)29-s + (0.433 − 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.651 + 0.758i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10665 - 0.508445i\)
\(L(\frac12)\) \(\approx\) \(1.10665 - 0.508445i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.82 + 3i)T \)
good3 \( 1 + (-1 + 0.414i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.29 + 3.12i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1 + 2.41i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-4.41 - 1.82i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.41iT - 13T^{2} \)
19 \( 1 + (-2.58 - 2.58i)T + 19iT^{2} \)
23 \( 1 + (0.414 + 0.171i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.878 + 2.12i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.41 + i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.121 + 0.0502i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.46 - 3.53i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (8.24 - 8.24i)T - 43iT^{2} \)
47 \( 1 + 6.82iT - 47T^{2} \)
53 \( 1 + (-0.171 - 0.171i)T + 53iT^{2} \)
59 \( 1 + (-9.07 + 9.07i)T - 59iT^{2} \)
61 \( 1 + (0.121 - 0.292i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 2.82T + 67T^{2} \)
71 \( 1 + (13.4 - 5.58i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.36 + 12.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.41 - 1.82i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-5.07 - 5.07i)T + 83iT^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + (-0.121 - 0.292i)T + (-68.5 + 68.5i)T^{2} \)
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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

−13.27328343957165144099297822553, −11.90114279156380337144886522783, −11.46099790709767508771749769329, −9.629695352347948756626295373758, −8.830346983208580652371693380573, −7.895129857534132766342874709101, −6.80536791675378667384829325341, −4.81675460955519326781157788721, −4.00941102580356670507514581890, −1.57482980005545839984243177946, 2.80719149161342445075862457227, 3.70721313816379295080794772486, 5.76075044310181780463299303144, 6.87419274007608039196708939197, 8.243175245549444512582356854906, 9.001935557050451030753754798881, 10.38076382238359406498280776106, 11.37996954486150314595464620618, 12.06153005348240939835384989790, 13.61006413885175266154597336495

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