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Properties

Label 2-136-17.15-c3-0-8
Degree $2$
Conductor $136$
Sign $-0.129 + 0.991i$
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  + (0.603 + 1.45i)3-s + (−10.3 + 4.29i)5-s + (−13.2 − 5.47i)7-s + (17.3 − 17.3i)9-s + (16.3 − 39.5i)11-s − 70.4i·13-s + (−12.5 − 12.5i)15-s + (−52.1 + 46.8i)17-s + (25.9 + 25.9i)19-s − 22.5i·21-s + (69.8 − 168. i)23-s + (0.842 − 0.842i)25-s + (75.0 + 31.0i)27-s + (−244. + 101. i)29-s + (−130. − 315. i)31-s + ⋯
L(s)  = 1  + (0.116 + 0.280i)3-s + (−0.928 + 0.384i)5-s + (−0.713 − 0.295i)7-s + (0.642 − 0.642i)9-s + (0.449 − 1.08i)11-s − 1.50i·13-s + (−0.215 − 0.215i)15-s + (−0.744 + 0.667i)17-s + (0.312 + 0.312i)19-s − 0.234i·21-s + (0.633 − 1.52i)23-s + (0.00674 − 0.00674i)25-s + (0.534 + 0.221i)27-s + (−1.56 + 0.648i)29-s + (−0.757 − 1.82i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.129 + 0.991i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.129 + 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.611161 - 0.696178i\)
\(L(\frac12)\) \(\approx\) \(0.611161 - 0.696178i\)
\(L(\frac{5}{2})\) 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 \( 1 + (52.1 - 46.8i)T \)
good3 \( 1 + (-0.603 - 1.45i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (10.3 - 4.29i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (13.2 + 5.47i)T + (242. + 242. i)T^{2} \)
11 \( 1 + (-16.3 + 39.5i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + 70.4iT - 2.19e3T^{2} \)
19 \( 1 + (-25.9 - 25.9i)T + 6.85e3iT^{2} \)
23 \( 1 + (-69.8 + 168. i)T + (-8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (244. - 101. i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (130. + 315. i)T + (-2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (-25.0 - 60.5i)T + (-3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (-13.9 - 5.79i)T + (4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (105. - 105. i)T - 7.95e4iT^{2} \)
47 \( 1 - 4.93iT - 1.03e5T^{2} \)
53 \( 1 + (-465. - 465. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-74.6 + 74.6i)T - 2.05e5iT^{2} \)
61 \( 1 + (543. + 225. i)T + (1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 952.T + 3.00e5T^{2} \)
71 \( 1 + (-127. - 307. i)T + (-2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (622. - 257. i)T + (2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-201. + 485. i)T + (-3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (-261. - 261. i)T + 5.71e5iT^{2} \)
89 \( 1 + 234. iT - 7.04e5T^{2} \)
97 \( 1 + (-124. + 51.6i)T + (6.45e5 - 6.45e5i)T^{2} \)
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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

−12.61991431361664052785518366584, −11.28020357553672929658723856456, −10.53237551396779386361749893041, −9.357797306709748617377686027350, −8.199278584045645242571573662134, −7.06471072207897694028523844211, −5.92080218602436813260003803948, −4.00950468734708596554545666179, −3.25544414888574037749824464095, −0.46192176494733000515730423585, 1.86873701681415399533054394417, 3.85172145580690973100466175822, 4.95128248220207634394375442422, 6.89418274325112143207845863758, 7.40962656591535551201473446972, 8.971566671026969373347404236271, 9.657047544148798332312554146058, 11.25808359361279536884741477833, 12.02525959115736845226104309139, 12.95087261366595356207402528010

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