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Properties

Label 2-136-136.115-c2-0-3
Degree $2$
Conductor $136$
Sign $0.881 - 0.472i$
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  + (−1.17 − 1.61i)2-s + (−3.41 − 3.41i)3-s + (−1.21 + 3.81i)4-s + (−2.66 + 2.66i)5-s + (−1.48 + 9.55i)6-s + (−1.35 − 1.35i)7-s + (7.58 − 2.53i)8-s + 14.3i·9-s + (7.44 + 1.15i)10-s + (−1.28 + 1.28i)11-s + (17.1 − 8.87i)12-s + 1.81i·13-s + (−0.591 + 3.79i)14-s + 18.2·15-s + (−13.0 − 9.26i)16-s + (12.5 + 11.4i)17-s + ⋯
L(s)  = 1  + (−0.589 − 0.807i)2-s + (−1.13 − 1.13i)3-s + (−0.303 + 0.952i)4-s + (−0.532 + 0.532i)5-s + (−0.247 + 1.59i)6-s + (−0.194 − 0.194i)7-s + (0.948 − 0.316i)8-s + 1.59i·9-s + (0.744 + 0.115i)10-s + (−0.116 + 0.116i)11-s + (1.43 − 0.739i)12-s + 0.139i·13-s + (−0.0422 + 0.271i)14-s + 1.21·15-s + (−0.815 − 0.579i)16-s + (0.739 + 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ 0.881 - 0.472i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.321709 + 0.0807467i\)
\(L(\frac12)\) \(\approx\) \(0.321709 + 0.0807467i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 1.61i)T \)
17 \( 1 + (-12.5 - 11.4i)T \)
good3 \( 1 + (3.41 + 3.41i)T + 9iT^{2} \)
5 \( 1 + (2.66 - 2.66i)T - 25iT^{2} \)
7 \( 1 + (1.35 + 1.35i)T + 49iT^{2} \)
11 \( 1 + (1.28 - 1.28i)T - 121iT^{2} \)
13 \( 1 - 1.81iT - 169T^{2} \)
19 \( 1 + 22.4iT - 361T^{2} \)
23 \( 1 + (-27.5 - 27.5i)T + 529iT^{2} \)
29 \( 1 + (30.5 - 30.5i)T - 841iT^{2} \)
31 \( 1 + (23.2 - 23.2i)T - 961iT^{2} \)
37 \( 1 + (26.0 - 26.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (9.47 - 9.47i)T - 1.68e3iT^{2} \)
43 \( 1 - 16.0iT - 1.84e3T^{2} \)
47 \( 1 + 25.0iT - 2.20e3T^{2} \)
53 \( 1 - 58.5T + 2.80e3T^{2} \)
59 \( 1 - 75.3iT - 3.48e3T^{2} \)
61 \( 1 + (16.8 + 16.8i)T + 3.72e3iT^{2} \)
67 \( 1 - 15.0T + 4.48e3T^{2} \)
71 \( 1 + (85.7 - 85.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (22.4 + 22.4i)T + 5.32e3iT^{2} \)
79 \( 1 + (98.8 + 98.8i)T + 6.24e3iT^{2} \)
83 \( 1 - 24.1iT - 6.88e3T^{2} \)
89 \( 1 + 39.1T + 7.92e3T^{2} \)
97 \( 1 + (-121. - 121. i)T + 9.40e3iT^{2} \)
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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

−12.85698547087414495405329119905, −11.79055851533822924428071704686, −11.22805813179215568373853974177, −10.39084352169694988793802459795, −8.893687514531307404378102593335, −7.35890357200008305818349723947, −7.04352381391838991390166665270, −5.28178728527440482432555513056, −3.35149282360727514552659778151, −1.39883065007373463985977993836, 0.33870444821100034103273149945, 4.12861742906176870256806945686, 5.23581814149348774733774642626, 6.04591096911256324046365326572, 7.55438955681084518473467285601, 8.826022195195658526466887549462, 9.799834457287881897598066182921, 10.63657179861491779201027307561, 11.61511100617910548665045182644, 12.70460540321978661556070820590

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