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Properties

Label 2-136-17.13-c3-0-10
Degree $2$
Conductor $136$
Sign $-0.531 + 0.847i$
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  + (−5.11 − 5.11i)3-s + (7.20 + 7.20i)5-s + (19.2 − 19.2i)7-s + 25.3i·9-s + (5.30 − 5.30i)11-s − 59.4·13-s − 73.6i·15-s + (−7.18 − 69.7i)17-s − 114. i·19-s − 196.·21-s + (−63.2 + 63.2i)23-s − 21.2i·25-s + (−8.60 + 8.60i)27-s + (−10.6 − 10.6i)29-s + (−143. − 143. i)31-s + ⋯
L(s)  = 1  + (−0.984 − 0.984i)3-s + (0.644 + 0.644i)5-s + (1.03 − 1.03i)7-s + 0.937i·9-s + (0.145 − 0.145i)11-s − 1.26·13-s − 1.26i·15-s + (−0.102 − 0.994i)17-s − 1.38i·19-s − 2.04·21-s + (−0.573 + 0.573i)23-s − 0.169i·25-s + (−0.0613 + 0.0613i)27-s + (−0.0685 − 0.0685i)29-s + (−0.830 − 0.830i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.531 + 0.847i$
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: \((2,\ 136,\ (\ :3/2),\ -0.531 + 0.847i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.550881 - 0.995801i\)
\(L(\frac12)\) \(\approx\) \(0.550881 - 0.995801i\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.18 + 69.7i)T \)
good3 \( 1 + (5.11 + 5.11i)T + 27iT^{2} \)
5 \( 1 + (-7.20 - 7.20i)T + 125iT^{2} \)
7 \( 1 + (-19.2 + 19.2i)T - 343iT^{2} \)
11 \( 1 + (-5.30 + 5.30i)T - 1.33e3iT^{2} \)
13 \( 1 + 59.4T + 2.19e3T^{2} \)
19 \( 1 + 114. iT - 6.85e3T^{2} \)
23 \( 1 + (63.2 - 63.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (10.6 + 10.6i)T + 2.43e4iT^{2} \)
31 \( 1 + (143. + 143. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-170. - 170. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-243. + 243. i)T - 6.89e4iT^{2} \)
43 \( 1 + 201. iT - 7.95e4T^{2} \)
47 \( 1 - 422.T + 1.03e5T^{2} \)
53 \( 1 - 538. iT - 1.48e5T^{2} \)
59 \( 1 + 330. iT - 2.05e5T^{2} \)
61 \( 1 + (404. - 404. i)T - 2.26e5iT^{2} \)
67 \( 1 - 40.2T + 3.00e5T^{2} \)
71 \( 1 + (694. + 694. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-541. - 541. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-585. + 585. i)T - 4.93e5iT^{2} \)
83 \( 1 - 670. iT - 5.71e5T^{2} \)
89 \( 1 + 130.T + 7.04e5T^{2} \)
97 \( 1 + (262. + 262. i)T + 9.12e5iT^{2} \)
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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

−12.25389525565832828464518650627, −11.37023896397537866173411377514, −10.67939352787540779151518203999, −9.433834115688711450489148834082, −7.49047770079488581213103413401, −7.13770898747972833583777629600, −5.84451014659945847569159261992, −4.63646863662470713901999470371, −2.26973876910910949652774272963, −0.64519619722214874298092010721, 1.90682843156780442814670767838, 4.35064645655194292111422158115, 5.30038207298862163602695601487, 5.95096294417065154361722973901, 7.955212258445440953961614410971, 9.151529342147656815847034823703, 10.03799574571453255612346274073, 11.01099294412329877012936267629, 12.06808463034629470515539065212, 12.67432281600481820856361987382

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