Properties

Label 2-136-136.19-c2-0-3
Degree $2$
Conductor $136$
Sign $-0.698 + 0.715i$
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  + (1.41 + 1.41i)2-s + (−5.53 + 2.29i)3-s + 4.00i·4-s + (−11.0 − 4.58i)6-s + (−5.65 + 5.65i)8-s + (19.0 − 19.0i)9-s + (−17.9 − 7.43i)11-s + (−9.17 − 22.1i)12-s − 16.0·16-s + (−11.2 + 12.7i)17-s + 53.7·18-s + (12 + 12i)19-s + (−14.8 − 35.8i)22-s + (18.3 − 44.2i)24-s + (−17.6 + 17.6i)25-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−1.84 + 0.764i)3-s + 1.00i·4-s + (−1.84 − 0.764i)6-s + (−0.707 + 0.707i)8-s + (2.11 − 2.11i)9-s + (−1.63 − 0.675i)11-s + (−0.764 − 1.84i)12-s − 1.00·16-s + (−0.664 + 0.747i)17-s + 2.98·18-s + (0.631 + 0.631i)19-s + (−0.675 − 1.63i)22-s + (0.764 − 1.84i)24-s + (−0.707 + 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ -0.698 + 0.715i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.178265 - 0.422977i\)
\(L(\frac12)\) \(\approx\) \(0.178265 - 0.422977i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 1.41i)T \)
17 \( 1 + (11.2 - 12.7i)T \)
good3 \( 1 + (5.53 - 2.29i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (34.6 + 34.6i)T^{2} \)
11 \( 1 + (17.9 + 7.43i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + 169T^{2} \)
19 \( 1 + (-12 - 12i)T + 361iT^{2} \)
23 \( 1 + (-374. - 374. i)T^{2} \)
29 \( 1 + (594. - 594. i)T^{2} \)
31 \( 1 + (-679. + 679. i)T^{2} \)
37 \( 1 + (-968. + 968. i)T^{2} \)
41 \( 1 + (6.32 - 15.2i)T + (-1.18e3 - 1.18e3i)T^{2} \)
43 \( 1 + (35.4 - 35.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 2.80e3iT^{2} \)
59 \( 1 + (1.42 - 1.42i)T - 3.48e3iT^{2} \)
61 \( 1 + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 - 127.T + 4.48e3T^{2} \)
71 \( 1 + (-3.56e3 + 3.56e3i)T^{2} \)
73 \( 1 + (-45.2 - 109. i)T + (-3.76e3 + 3.76e3i)T^{2} \)
79 \( 1 + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 + (-53.5 - 53.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 31.2iT - 7.92e3T^{2} \)
97 \( 1 + (-73.7 - 178. i)T + (-6.65e3 + 6.65e3i)T^{2} \)
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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

−13.35300791159090642660979987185, −12.63743027303996130071604918099, −11.55201913914796721439757309100, −10.85791613807724297203387355832, −9.749222850149932454411201964159, −8.037503672281955575631037346500, −6.66412501359125649282107588343, −5.66559296558339253301988464031, −5.03374573111986217646359261119, −3.69997783574079313346639959300, 0.29118608353256874582739739615, 2.18670113939316599113085848528, 4.73050040326917677671917785575, 5.34064946866851208897943966489, 6.54249291934765986263438840273, 7.55799808205574953853409759381, 9.882775110283980103666289559746, 10.68206573622284675144707120197, 11.48934549146640269775518857829, 12.27859599641287913964018148577

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