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Properties

Label 2-136-136.59-c0-0-0
Degree $2$
Conductor $136$
Sign $0.997 - 0.0758i$
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 + 1.70i)3-s − 1.00i·4-s + (0.707 + 1.70i)6-s + (−0.707 − 0.707i)8-s + (−1.70 − 1.70i)9-s + (−0.292 − 0.707i)11-s + (1.70 + 0.707i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 2.41·18-s + (−0.707 − 0.292i)22-s + (1.70 − 0.707i)24-s + (0.707 + 0.707i)25-s + (2.41 − i)27-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 + 1.70i)3-s − 1.00i·4-s + (0.707 + 1.70i)6-s + (−0.707 − 0.707i)8-s + (−1.70 − 1.70i)9-s + (−0.292 − 0.707i)11-s + (1.70 + 0.707i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 2.41·18-s + (−0.707 − 0.292i)22-s + (1.70 − 0.707i)24-s + (0.707 + 0.707i)25-s + (2.41 − i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(0.0678728\)
Root analytic conductor: \(0.260524\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :0),\ 0.997 - 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6829298843\)
\(L(\frac12)\) \(\approx\) \(0.6829298843\)
\(L(1)\) 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 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
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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

−13.44368421954882884825218177662, −12.22899658041228817460497100220, −11.10916997911299448449327189141, −10.76393502517288503749814816811, −9.718940012970763598304128826386, −8.779608712820201062561950996831, −6.30848864215535830980669951229, −5.33526846793379661168496317593, −4.33544363119264524956986759978, −3.21670837177715010608350948468, 2.45018497680460612401004236085, 4.80109863993334326261410933505, 5.96652199153184866586418734347, 6.94952358543983213480756466490, 7.58372770452251988614410918593, 8.776340600704764084919352485455, 10.85949231581725531108381428441, 11.99696042546515032047883231250, 12.49083967670888564397425605649, 13.43530141014583339936138890903

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