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Properties

Label 2-136-8.5-c1-0-7
Degree $2$
Conductor $136$
Sign $0.335 - 0.941i$
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  + (1.13 + 0.842i)2-s + 0.750i·3-s + (0.581 + 1.91i)4-s − 0.436i·5-s + (−0.632 + 0.852i)6-s − 1.90·7-s + (−0.949 + 2.66i)8-s + 2.43·9-s + (0.367 − 0.496i)10-s − 2.45i·11-s + (−1.43 + 0.436i)12-s − 2.93i·13-s + (−2.16 − 1.60i)14-s + 0.327·15-s + (−3.32 + 2.22i)16-s − 17-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + 0.433i·3-s + (0.290 + 0.956i)4-s − 0.195i·5-s + (−0.258 + 0.348i)6-s − 0.719·7-s + (−0.335 + 0.941i)8-s + 0.812·9-s + (0.116 − 0.156i)10-s − 0.740i·11-s + (−0.414 + 0.126i)12-s − 0.812i·13-s + (−0.578 − 0.428i)14-s + 0.0846·15-s + (−0.830 + 0.556i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.335 - 0.941i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.335 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28391 + 0.905312i\)
\(L(\frac12)\) \(\approx\) \(1.28391 + 0.905312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.842i)T \)
17 \( 1 + T \)
good3 \( 1 - 0.750iT - 3T^{2} \)
5 \( 1 + 0.436iT - 5T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 + 2.45iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
19 \( 1 + 0.713iT - 19T^{2} \)
23 \( 1 + 0.640T + 23T^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 + 6.23T + 31T^{2} \)
37 \( 1 - 1.93iT - 37T^{2} \)
41 \( 1 + 1.06T + 41T^{2} \)
43 \( 1 - 7.65iT - 43T^{2} \)
47 \( 1 + 2.73T + 47T^{2} \)
53 \( 1 - 4.91iT - 53T^{2} \)
59 \( 1 + 6.77iT - 59T^{2} \)
61 \( 1 - 13.9iT - 61T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + 9.52T + 73T^{2} \)
79 \( 1 + 3.35T + 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 - 0.155T + 89T^{2} \)
97 \( 1 - 3.59T + 97T^{2} \)
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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

−13.17374066630305996944883383603, −12.89479959070934429861331136143, −11.55443750416483334869157913841, −10.38871636023041057950684840377, −9.160929979613611134195008673607, −7.948000969985091147032607060326, −6.73676763416298057454889011473, −5.61922380939174426732883687488, −4.33861836177733798554049239639, −3.10872361941836952711581112882, 1.90409121388229592781889431955, 3.61193516096937532450765325389, 4.92709872239717786377274104443, 6.53699823635990515215726843833, 7.15383295620745841851915024294, 9.177600342671128357819389013649, 10.10556501458144295425794053966, 11.10752484025849527323340198879, 12.43319398363684204055279712135, 12.74213006196894102061772929464

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