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Properties

Label 2-136-17.9-c3-0-8
Degree $2$
Conductor $136$
Sign $0.890 - 0.454i$
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  + (8.27 + 3.42i)3-s + (1.41 − 3.41i)5-s + (−0.563 − 1.36i)7-s + (37.6 + 37.6i)9-s + (54.6 − 22.6i)11-s + 12.0i·13-s + (23.3 − 23.3i)15-s + (−63.8 + 28.9i)17-s + (−29.8 + 29.8i)19-s − 13.1i·21-s + (−91.5 + 37.9i)23-s + (78.7 + 78.7i)25-s + (90.0 + 217. i)27-s + (97.5 − 235. i)29-s + (−155. − 64.5i)31-s + ⋯
L(s)  = 1  + (1.59 + 0.659i)3-s + (0.126 − 0.305i)5-s + (−0.0304 − 0.0734i)7-s + (1.39 + 1.39i)9-s + (1.49 − 0.620i)11-s + 0.257i·13-s + (0.402 − 0.402i)15-s + (−0.910 + 0.413i)17-s + (−0.360 + 0.360i)19-s − 0.137i·21-s + (−0.829 + 0.343i)23-s + (0.629 + 0.629i)25-s + (0.642 + 1.55i)27-s + (0.624 − 1.50i)29-s + (−0.903 − 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.76283 + 0.664846i\)
\(L(\frac12)\) \(\approx\) \(2.76283 + 0.664846i\)
\(L(\frac{5}{2})\) 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 \( 1 + (63.8 - 28.9i)T \)
good3 \( 1 + (-8.27 - 3.42i)T + (19.0 + 19.0i)T^{2} \)
5 \( 1 + (-1.41 + 3.41i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (0.563 + 1.36i)T + (-242. + 242. i)T^{2} \)
11 \( 1 + (-54.6 + 22.6i)T + (941. - 941. i)T^{2} \)
13 \( 1 - 12.0iT - 2.19e3T^{2} \)
19 \( 1 + (29.8 - 29.8i)T - 6.85e3iT^{2} \)
23 \( 1 + (91.5 - 37.9i)T + (8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (-97.5 + 235. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (155. + 64.5i)T + (2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (70.9 + 29.3i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-18.0 - 43.5i)T + (-4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (160. + 160. i)T + 7.95e4iT^{2} \)
47 \( 1 - 279. iT - 1.03e5T^{2} \)
53 \( 1 + (149. - 149. i)T - 1.48e5iT^{2} \)
59 \( 1 + (74.5 + 74.5i)T + 2.05e5iT^{2} \)
61 \( 1 + (316. + 763. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + 638.T + 3.00e5T^{2} \)
71 \( 1 + (998. + 413. i)T + (2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (54.5 - 131. i)T + (-2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-201. + 83.5i)T + (3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (-883. + 883. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.22e3iT - 7.04e5T^{2} \)
97 \( 1 + (-66.0 + 159. i)T + (-6.45e5 - 6.45e5i)T^{2} \)
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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

−13.20502261165679538181492056544, −11.80748094629664285651979188684, −10.51547690282771894755642147134, −9.316325633936317560278652970944, −8.882724852891469874639099390320, −7.82589615704573877125624741371, −6.31253122800649645168967111386, −4.38140713455173487092718738816, −3.53668594330003550299124755597, −1.89107731094651368842411071319, 1.66196697341950403136447114777, 2.94077269203782435481902931782, 4.29063459103099662050278356005, 6.57406523641069487627515576302, 7.24000635626940995761270988501, 8.651114959647365775957796157739, 9.132946346234672787727792412096, 10.40528351122980092587849984924, 11.96555850760396292914586567445, 12.82094131406798306824906827034

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