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Properties

Label 2-136-136.59-c2-0-21
Degree $2$
Conductor $136$
Sign $0.134 + 0.990i$
Analytic cond. $3.70573$
Root an. cond. $1.92502$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)2-s + (1.53 − 3.70i)3-s − 4.00i·4-s + (3.07 + 7.41i)6-s + (5.65 + 5.65i)8-s + (−5.02 − 5.02i)9-s + (−8.05 − 19.4i)11-s + (−14.8 − 6.14i)12-s − 16.0·16-s + (−12.7 − 11.2i)17-s + 14.2·18-s + (12 − 12i)19-s + (38.8 + 16.1i)22-s + (29.6 − 12.2i)24-s + (17.6 + 17.6i)25-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.511 − 1.23i)3-s − 1.00i·4-s + (0.511 + 1.23i)6-s + (0.707 + 0.707i)8-s + (−0.557 − 0.557i)9-s + (−0.731 − 1.76i)11-s + (−1.23 − 0.511i)12-s − 1.00·16-s + (−0.747 − 0.664i)17-s + 0.788·18-s + (0.631 − 0.631i)19-s + (1.76 + 0.731i)22-s + (1.23 − 0.511i)24-s + (0.707 + 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.134 + 0.990i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ 0.134 + 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.776620 - 0.678391i\)
\(L(\frac12)\) \(\approx\) \(0.776620 - 0.678391i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 1.41i)T \)
17 \( 1 + (12.7 + 11.2i)T \)
good3 \( 1 + (-1.53 + 3.70i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-34.6 + 34.6i)T^{2} \)
11 \( 1 + (8.05 + 19.4i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + 169T^{2} \)
19 \( 1 + (-12 + 12i)T - 361iT^{2} \)
23 \( 1 + (374. - 374. i)T^{2} \)
29 \( 1 + (-594. - 594. i)T^{2} \)
31 \( 1 + (679. + 679. i)T^{2} \)
37 \( 1 + (968. + 968. i)T^{2} \)
41 \( 1 + (41.6 - 17.2i)T + (1.18e3 - 1.18e3i)T^{2} \)
43 \( 1 + (-49.4 - 49.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3iT^{2} \)
59 \( 1 + (-83.4 - 83.4i)T + 3.48e3iT^{2} \)
61 \( 1 + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 - 40.1T + 4.48e3T^{2} \)
71 \( 1 + (3.56e3 + 3.56e3i)T^{2} \)
73 \( 1 + (21.2 + 8.79i)T + (3.76e3 + 3.76e3i)T^{2} \)
79 \( 1 + (4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (-104. + 104. i)T - 6.88e3iT^{2} \)
89 \( 1 - 175. iT - 7.92e3T^{2} \)
97 \( 1 + (-140. - 58.0i)T + (6.65e3 + 6.65e3i)T^{2} \)
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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

−13.32002804633154634985981182723, −11.59768264600216652564138266588, −10.67928895415685122050452350036, −9.135317368552426831272483295389, −8.363421763086942462675279237122, −7.45533150013843930075509543139, −6.53044857902942224821520901712, −5.29388460417709564104902002898, −2.70382030510955332830316749314, −0.843286311765060617672073876087, 2.29330489815838369534410705368, 3.81398568526402610069685642336, 4.82484061771453811918804069193, 7.10181623202252572917931738782, 8.311695761603839241481703169737, 9.318967917415573251398136815639, 10.13912152639144287054422105259, 10.66814874514002142945376220312, 12.11624964883503568392313683026, 12.93022850862517217068524184796

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