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Properties

Label 2-136-136.43-c0-0-0
Degree $2$
Conductor $136$
Sign $0.673 - 0.739i$
Analytic cond. $0.0678728$
Root an. cond. $0.260524$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.292i)3-s − 1.00i·4-s + (−0.707 + 0.292i)6-s + (0.707 + 0.707i)8-s + (−0.292 − 0.292i)9-s + (−1.70 + 0.707i)11-s + (0.292 − 0.707i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + 0.414·18-s + (0.707 − 1.70i)22-s + (0.292 + 0.707i)24-s + (−0.707 − 0.707i)25-s + (−0.414 − 1.00i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.292i)3-s − 1.00i·4-s + (−0.707 + 0.292i)6-s + (0.707 + 0.707i)8-s + (−0.292 − 0.292i)9-s + (−1.70 + 0.707i)11-s + (0.292 − 0.707i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + 0.414·18-s + (0.707 − 1.70i)22-s + (0.292 + 0.707i)24-s + (−0.707 − 0.707i)25-s + (−0.414 − 1.00i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(0.0678728\)
Root analytic conductor: \(0.260524\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :0),\ 0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5198748038\)
\(L(\frac12)\) \(\approx\) \(0.5198748038\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (1 + i)T + iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
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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

−13.97413905032126940313467226696, −12.77904963378803838800436013130, −11.29691684278355051237684496577, −10.05592689375429114078977474895, −9.483287034266436882843707780577, −8.175611521551417626168374837597, −7.58492444758015215348649087670, −6.03618322896740243925109865513, −4.76271806022856407544235163714, −2.66511094688412949363541073762, 2.29087843910759297804616113393, 3.48611060907769416643412290216, 5.50596068933523817240724901115, 7.52382658591326990188196679549, 8.107939694813905132159077763354, 9.069770138442056382190040664788, 10.37174911285377785543044451093, 11.03418865132874643147833381902, 12.36262238608768568123969708863, 13.26340828794982565881175379298

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