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Properties

Label 2-136-17.16-c3-0-11
Degree $2$
Conductor $136$
Sign $-0.957 + 0.290i$
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  − 8.27i·3-s + 6.42i·5-s − 24.9i·7-s − 41.4·9-s − 18.4i·11-s + 10.7·13-s + 53.1·15-s + (−67.0 + 20.3i)17-s − 130.·19-s − 206.·21-s + 90.2i·23-s + 83.6·25-s + 119. i·27-s + 29.6i·29-s − 132. i·31-s + ⋯
L(s)  = 1  − 1.59i·3-s + 0.574i·5-s − 1.34i·7-s − 1.53·9-s − 0.506i·11-s + 0.229·13-s + 0.915·15-s + (−0.957 + 0.290i)17-s − 1.56·19-s − 2.14·21-s + 0.818i·23-s + 0.669·25-s + 0.849i·27-s + 0.189i·29-s − 0.769i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.957 + 0.290i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.957 + 0.290i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.174506 - 1.17744i\)
\(L(\frac12)\) \(\approx\) \(0.174506 - 1.17744i\)
\(L(\frac{5}{2})\) 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 \( 1 + (67.0 - 20.3i)T \)
good3 \( 1 + 8.27iT - 27T^{2} \)
5 \( 1 - 6.42iT - 125T^{2} \)
7 \( 1 + 24.9iT - 343T^{2} \)
11 \( 1 + 18.4iT - 1.33e3T^{2} \)
13 \( 1 - 10.7T + 2.19e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 - 90.2iT - 1.21e4T^{2} \)
29 \( 1 - 29.6iT - 2.43e4T^{2} \)
31 \( 1 + 132. iT - 2.97e4T^{2} \)
37 \( 1 + 412. iT - 5.06e4T^{2} \)
41 \( 1 + 165. iT - 6.89e4T^{2} \)
43 \( 1 - 502.T + 7.95e4T^{2} \)
47 \( 1 + 160.T + 1.03e5T^{2} \)
53 \( 1 - 372.T + 1.48e5T^{2} \)
59 \( 1 - 453.T + 2.05e5T^{2} \)
61 \( 1 + 254. iT - 2.26e5T^{2} \)
67 \( 1 + 383.T + 3.00e5T^{2} \)
71 \( 1 + 818. iT - 3.57e5T^{2} \)
73 \( 1 - 640. iT - 3.89e5T^{2} \)
79 \( 1 + 380. iT - 4.93e5T^{2} \)
83 \( 1 - 984.T + 5.71e5T^{2} \)
89 \( 1 - 417.T + 7.04e5T^{2} \)
97 \( 1 - 1.32e3iT - 9.12e5T^{2} \)
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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

−12.56802514182460113998589134437, −11.15772773044001934054071486601, −10.64261264482492029020116748744, −8.832811951588462673941653458680, −7.66050790302490712814232669363, −6.93520921865436682592225967636, −6.07694502261724614722633975607, −3.94783025105502706965747629658, −2.18405244704996235070322980842, −0.58453120433780344524161841525, 2.57595863097080350482764679078, 4.31650041282517715993619703463, 5.05752258620027300644014952070, 6.36249169660863205093772403905, 8.619795919843707127640449226840, 8.902719233327988237349351222444, 10.06262954360932385762988967381, 11.00475748218160905314457769248, 12.10276870676141076789080191000, 13.06057007892849318460748382585

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