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Properties

Label 2-136-17.13-c3-0-12
Degree $2$
Conductor $136$
Sign $-0.850 + 0.526i$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.13 + 3.13i)3-s + (−13.6 − 13.6i)5-s + (−0.366 + 0.366i)7-s − 7.36i·9-s + (−48.7 + 48.7i)11-s − 78.1·13-s − 85.8i·15-s + (24.2 − 65.7i)17-s − 20.0i·19-s − 2.29·21-s + (76.8 − 76.8i)23-s + 250. i·25-s + (107. − 107. i)27-s + (−69.3 − 69.3i)29-s + (−181. − 181. i)31-s + ⋯
L(s)  = 1  + (0.602 + 0.602i)3-s + (−1.22 − 1.22i)5-s + (−0.0197 + 0.0197i)7-s − 0.272i·9-s + (−1.33 + 1.33i)11-s − 1.66·13-s − 1.47i·15-s + (0.346 − 0.938i)17-s − 0.242i·19-s − 0.0238·21-s + (0.696 − 0.696i)23-s + 2.00i·25-s + (0.767 − 0.767i)27-s + (−0.444 − 0.444i)29-s + (−1.05 − 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.850 + 0.526i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.850 + 0.526i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.114474 - 0.402709i\)
\(L(\frac12)\) \(\approx\) \(0.114474 - 0.402709i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-24.2 + 65.7i)T \)
good3 \( 1 + (-3.13 - 3.13i)T + 27iT^{2} \)
5 \( 1 + (13.6 + 13.6i)T + 125iT^{2} \)
7 \( 1 + (0.366 - 0.366i)T - 343iT^{2} \)
11 \( 1 + (48.7 - 48.7i)T - 1.33e3iT^{2} \)
13 \( 1 + 78.1T + 2.19e3T^{2} \)
19 \( 1 + 20.0iT - 6.85e3T^{2} \)
23 \( 1 + (-76.8 + 76.8i)T - 1.21e4iT^{2} \)
29 \( 1 + (69.3 + 69.3i)T + 2.43e4iT^{2} \)
31 \( 1 + (181. + 181. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-209. - 209. i)T + 5.06e4iT^{2} \)
41 \( 1 + (43.1 - 43.1i)T - 6.89e4iT^{2} \)
43 \( 1 - 494. iT - 7.95e4T^{2} \)
47 \( 1 - 8.61T + 1.03e5T^{2} \)
53 \( 1 + 484. iT - 1.48e5T^{2} \)
59 \( 1 + 229. iT - 2.05e5T^{2} \)
61 \( 1 + (69.4 - 69.4i)T - 2.26e5iT^{2} \)
67 \( 1 - 42.5T + 3.00e5T^{2} \)
71 \( 1 + (135. + 135. i)T + 3.57e5iT^{2} \)
73 \( 1 + (579. + 579. i)T + 3.89e5iT^{2} \)
79 \( 1 + (99.3 - 99.3i)T - 4.93e5iT^{2} \)
83 \( 1 - 108. iT - 5.71e5T^{2} \)
89 \( 1 - 150.T + 7.04e5T^{2} \)
97 \( 1 + (705. + 705. i)T + 9.12e5iT^{2} \)
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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

−12.45796327946328157852221859573, −11.49676952892534714506752745820, −9.867054264920756377121682258063, −9.326314216952746772838112569586, −7.997336673568669140024536633611, −7.36629343523391593544121763955, −4.94111102197280836667499493895, −4.47125653629092226469450000456, −2.77035993901261932955290089506, −0.18353609916501335495969979970, 2.54620095588783616186648772005, 3.49795542750078333378888049031, 5.41685468749687521574884148864, 7.23721634323167717501331395682, 7.59408128993996582461945605433, 8.597861386264009437847514846762, 10.43422669959293849338640864649, 10.97837680457518330002265329078, 12.21278888792582632236959476531, 13.17613113984412589237163885970

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