Properties

Label 2-136-17.10-c2-0-1
Degree $2$
Conductor $136$
Sign $0.406 - 0.913i$
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  + (−3.91 − 2.61i)3-s + (−7.19 + 1.43i)5-s + (10.7 + 2.13i)7-s + (5.02 + 12.1i)9-s + (6.65 + 9.96i)11-s + (−5.75 + 5.75i)13-s + (31.8 + 13.2i)15-s + (−3.01 + 16.7i)17-s + (8.02 − 19.3i)19-s + (−36.3 − 36.3i)21-s + (−21.5 + 14.3i)23-s + (26.6 − 11.0i)25-s + (3.79 − 19.0i)27-s + (9.12 + 45.8i)29-s + (−13.4 + 20.1i)31-s + ⋯
L(s)  = 1  + (−1.30 − 0.871i)3-s + (−1.43 + 0.286i)5-s + (1.53 + 0.304i)7-s + (0.558 + 1.34i)9-s + (0.605 + 0.906i)11-s + (−0.442 + 0.442i)13-s + (2.12 + 0.880i)15-s + (−0.177 + 0.984i)17-s + (0.422 − 1.01i)19-s + (−1.73 − 1.73i)21-s + (−0.935 + 0.625i)23-s + (1.06 − 0.440i)25-s + (0.140 − 0.707i)27-s + (0.314 + 1.58i)29-s + (−0.434 + 0.651i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.406 - 0.913i$
Analytic conductor: \(3.70573\)
Root analytic conductor: \(1.92502\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1),\ 0.406 - 0.913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.514707 + 0.334322i\)
\(L(\frac12)\) \(\approx\) \(0.514707 + 0.334322i\)
\(L(2)\) 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 \( 1 + (3.01 - 16.7i)T \)
good3 \( 1 + (3.91 + 2.61i)T + (3.44 + 8.31i)T^{2} \)
5 \( 1 + (7.19 - 1.43i)T + (23.0 - 9.56i)T^{2} \)
7 \( 1 + (-10.7 - 2.13i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (-6.65 - 9.96i)T + (-46.3 + 111. i)T^{2} \)
13 \( 1 + (5.75 - 5.75i)T - 169iT^{2} \)
19 \( 1 + (-8.02 + 19.3i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (21.5 - 14.3i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (-9.12 - 45.8i)T + (-776. + 321. i)T^{2} \)
31 \( 1 + (13.4 - 20.1i)T + (-367. - 887. i)T^{2} \)
37 \( 1 + (9.99 + 6.67i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (6.50 + 1.29i)T + (1.55e3 + 643. i)T^{2} \)
43 \( 1 + (-1.30 - 3.15i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (-37.6 + 37.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (20.3 - 49.2i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-80.9 + 33.5i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (0.981 - 4.93i)T + (-3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 - 78.4iT - 4.48e3T^{2} \)
71 \( 1 + (47.8 + 31.9i)T + (1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (97.5 - 19.3i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (-55.4 - 82.9i)T + (-2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (116. + 48.4i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (38.6 + 38.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (5.58 + 28.1i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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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

−12.65511228904723177836551641764, −11.87405543638500959229461499553, −11.52387042956297221442051704519, −10.63699352184122373674138210111, −8.656418259501443451524998432590, −7.47411702980636239798416592015, −6.93353475053980840160453658365, −5.30258258507162024187568745072, −4.26630161426575978555540592760, −1.58765910025271745477724723283, 0.52190327289559134308167366167, 3.97223786851639920193952287460, 4.69258610069471334550690549037, 5.83356618506198096396126220949, 7.56649833479797110378093322763, 8.398351112709443999131805122636, 9.998799890926534744475787686980, 11.11463935678768652541193921966, 11.62431801811808682140779026210, 12.11870393782791096869667483393

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