Properties

Label 2-136-17.12-c2-0-7
Degree $2$
Conductor $136$
Sign $0.929 + 0.368i$
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  + (3.29 − 2.19i)3-s + (8.73 + 1.73i)5-s + (−3.91 + 0.777i)7-s + (2.55 − 6.16i)9-s + (−8.01 + 11.9i)11-s + (−11.0 − 11.0i)13-s + (32.5 − 13.4i)15-s + (13.8 + 9.85i)17-s + (−10.4 − 25.1i)19-s + (−11.1 + 11.1i)21-s + (−21.8 − 14.6i)23-s + (50.1 + 20.7i)25-s + (1.79 + 9.02i)27-s + (2.53 − 12.7i)29-s + (2.57 + 3.84i)31-s + ⋯
L(s)  = 1  + (1.09 − 0.733i)3-s + (1.74 + 0.347i)5-s + (−0.558 + 0.111i)7-s + (0.283 − 0.685i)9-s + (−0.728 + 1.08i)11-s + (−0.851 − 0.851i)13-s + (2.17 − 0.899i)15-s + (0.815 + 0.579i)17-s + (−0.547 − 1.32i)19-s + (−0.531 + 0.531i)21-s + (−0.950 − 0.634i)23-s + (2.00 + 0.830i)25-s + (0.0665 + 0.334i)27-s + (0.0874 − 0.439i)29-s + (0.0829 + 0.124i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.19084 - 0.418552i\)
\(L(\frac12)\) \(\approx\) \(2.19084 - 0.418552i\)
\(L(2)\) 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 \( 1 + (-13.8 - 9.85i)T \)
good3 \( 1 + (-3.29 + 2.19i)T + (3.44 - 8.31i)T^{2} \)
5 \( 1 + (-8.73 - 1.73i)T + (23.0 + 9.56i)T^{2} \)
7 \( 1 + (3.91 - 0.777i)T + (45.2 - 18.7i)T^{2} \)
11 \( 1 + (8.01 - 11.9i)T + (-46.3 - 111. i)T^{2} \)
13 \( 1 + (11.0 + 11.0i)T + 169iT^{2} \)
19 \( 1 + (10.4 + 25.1i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (21.8 + 14.6i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (-2.53 + 12.7i)T + (-776. - 321. i)T^{2} \)
31 \( 1 + (-2.57 - 3.84i)T + (-367. + 887. i)T^{2} \)
37 \( 1 + (41.9 - 28.0i)T + (523. - 1.26e3i)T^{2} \)
41 \( 1 + (-2.05 + 0.408i)T + (1.55e3 - 643. i)T^{2} \)
43 \( 1 + (-14.1 + 34.2i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (9.45 + 9.45i)T + 2.20e3iT^{2} \)
53 \( 1 + (-14.7 - 35.6i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-34.6 - 14.3i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (4.97 + 25.0i)T + (-3.43e3 + 1.42e3i)T^{2} \)
67 \( 1 - 95.9iT - 4.48e3T^{2} \)
71 \( 1 + (-2.58 + 1.72i)T + (1.92e3 - 4.65e3i)T^{2} \)
73 \( 1 + (-92.3 - 18.3i)T + (4.92e3 + 2.03e3i)T^{2} \)
79 \( 1 + (37.2 - 55.7i)T + (-2.38e3 - 5.76e3i)T^{2} \)
83 \( 1 + (-69.6 + 28.8i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-98.9 + 98.9i)T - 7.92e3iT^{2} \)
97 \( 1 + (-7.35 + 36.9i)T + (-8.69e3 - 3.60e3i)T^{2} \)
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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

−12.99161609877434889772912920272, −12.53037124349525861493824054477, −10.32182106246109068134372572361, −9.913457506416114361282179539065, −8.749735597141148420869084198582, −7.51539112051142708562587348109, −6.51215831389021657501150832914, −5.21850329676407473541621894797, −2.79804807600216701641875608904, −2.08825530835950722465954341907, 2.16783295411362435577386188615, 3.47187778471055086315029500373, 5.20754550911405032278663518759, 6.26945008297252817622735216435, 8.033840066447436056962241403474, 9.177423166595563201595508202255, 9.757106864061519393760858744802, 10.44281277513646992427897674630, 12.26827166620135792510900302171, 13.41424003913366887584971266571

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