Properties

Label 2-136-17.6-c2-0-7
Degree $2$
Conductor $136$
Sign $0.404 + 0.914i$
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  + (1.35 − 0.269i)3-s + (4.43 − 6.64i)5-s + (−5.05 − 7.55i)7-s + (−6.55 + 2.71i)9-s + (−1.51 + 7.62i)11-s + (9.47 − 9.47i)13-s + (4.22 − 10.2i)15-s + (16.6 − 3.53i)17-s + (20.4 + 8.48i)19-s + (−8.88 − 8.88i)21-s + (−20.1 − 4.01i)23-s + (−14.8 − 35.8i)25-s + (−18.4 + 12.3i)27-s + (24.2 + 16.2i)29-s + (10.8 + 54.6i)31-s + ⋯
L(s)  = 1  + (0.451 − 0.0898i)3-s + (0.887 − 1.32i)5-s + (−0.721 − 1.07i)7-s + (−0.727 + 0.301i)9-s + (−0.137 + 0.693i)11-s + (0.729 − 0.729i)13-s + (0.281 − 0.680i)15-s + (0.978 − 0.207i)17-s + (1.07 + 0.446i)19-s + (−0.422 − 0.422i)21-s + (−0.878 − 0.174i)23-s + (−0.594 − 1.43i)25-s + (−0.684 + 0.457i)27-s + (0.836 + 0.559i)29-s + (0.350 + 1.76i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.404 + 0.914i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ 0.404 + 0.914i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.39671 - 0.909922i\)
\(L(\frac12)\) \(\approx\) \(1.39671 - 0.909922i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-16.6 + 3.53i)T \)
good3 \( 1 + (-1.35 + 0.269i)T + (8.31 - 3.44i)T^{2} \)
5 \( 1 + (-4.43 + 6.64i)T + (-9.56 - 23.0i)T^{2} \)
7 \( 1 + (5.05 + 7.55i)T + (-18.7 + 45.2i)T^{2} \)
11 \( 1 + (1.51 - 7.62i)T + (-111. - 46.3i)T^{2} \)
13 \( 1 + (-9.47 + 9.47i)T - 169iT^{2} \)
19 \( 1 + (-20.4 - 8.48i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (20.1 + 4.01i)T + (488. + 202. i)T^{2} \)
29 \( 1 + (-24.2 - 16.2i)T + (321. + 776. i)T^{2} \)
31 \( 1 + (-10.8 - 54.6i)T + (-887. + 367. i)T^{2} \)
37 \( 1 + (-43.1 + 8.57i)T + (1.26e3 - 523. i)T^{2} \)
41 \( 1 + (-9.07 - 13.5i)T + (-643. + 1.55e3i)T^{2} \)
43 \( 1 + (39.9 - 16.5i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (-23.2 + 23.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (53.2 + 22.0i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (3.95 + 9.55i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (88.8 - 59.3i)T + (1.42e3 - 3.43e3i)T^{2} \)
67 \( 1 + 1.75iT - 4.48e3T^{2} \)
71 \( 1 + (-21.6 + 4.31i)T + (4.65e3 - 1.92e3i)T^{2} \)
73 \( 1 + (64.9 - 97.1i)T + (-2.03e3 - 4.92e3i)T^{2} \)
79 \( 1 + (-7.36 + 37.0i)T + (-5.76e3 - 2.38e3i)T^{2} \)
83 \( 1 + (-58.6 + 141. i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-24.6 - 24.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (-50.5 - 33.7i)T + (3.60e3 + 8.69e3i)T^{2} \)
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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

−12.98446322570804644376079499897, −12.08393340794606211950140506085, −10.37370248658889507593718523127, −9.726316280579749557579047186352, −8.597720570689981351272546536601, −7.62421332556526246789901609948, −6.02227162307270645522324831211, −4.92822200130855694780049356001, −3.24693835987517475254195561497, −1.20550443260545090460422785078, 2.52566609600951850906070310413, 3.35283004427707397195323513813, 5.95736491411320657904816528221, 6.18108722700545564209414302344, 7.928166386511726428043685617550, 9.236658361392882337926038281775, 9.831644127114698942020344682742, 11.16686707369601189652455859495, 11.99655488628536724602628485359, 13.60191285050111416387745380710

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