Properties

Label 2-136-1.1-c3-0-0
Degree $2$
Conductor $136$
Sign $1$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.80·3-s − 18.3·5-s − 24.4·7-s + 69.1·9-s + 2.36·11-s + 17.2·13-s + 179.·15-s − 17·17-s − 136.·19-s + 239.·21-s + 2.04·23-s + 210.·25-s − 413.·27-s + 170.·29-s + 188.·31-s − 23.2·33-s + 447.·35-s − 137.·37-s − 169.·39-s − 260.·41-s − 502.·43-s − 1.26e3·45-s + 199.·47-s + 253.·49-s + 166.·51-s + 128.·53-s − 43.3·55-s + ⋯
L(s)  = 1  − 1.88·3-s − 1.63·5-s − 1.31·7-s + 2.56·9-s + 0.0649·11-s + 0.367·13-s + 3.09·15-s − 0.242·17-s − 1.65·19-s + 2.48·21-s + 0.0185·23-s + 1.68·25-s − 2.94·27-s + 1.09·29-s + 1.09·31-s − 0.122·33-s + 2.15·35-s − 0.612·37-s − 0.694·39-s − 0.992·41-s − 1.78·43-s − 4.19·45-s + 0.619·47-s + 0.737·49-s + 0.457·51-s + 0.333·53-s − 0.106·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $1$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2725396161\)
\(L(\frac12)\) \(\approx\) \(0.2725396161\)
\(L(\frac{5}{2})\) 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 \( 1 + 17T \)
good3 \( 1 + 9.80T + 27T^{2} \)
5 \( 1 + 18.3T + 125T^{2} \)
7 \( 1 + 24.4T + 343T^{2} \)
11 \( 1 - 2.36T + 1.33e3T^{2} \)
13 \( 1 - 17.2T + 2.19e3T^{2} \)
19 \( 1 + 136.T + 6.85e3T^{2} \)
23 \( 1 - 2.04T + 1.21e4T^{2} \)
29 \( 1 - 170.T + 2.43e4T^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 + 137.T + 5.06e4T^{2} \)
41 \( 1 + 260.T + 6.89e4T^{2} \)
43 \( 1 + 502.T + 7.95e4T^{2} \)
47 \( 1 - 199.T + 1.03e5T^{2} \)
53 \( 1 - 128.T + 1.48e5T^{2} \)
59 \( 1 - 303.T + 2.05e5T^{2} \)
61 \( 1 + 351.T + 2.26e5T^{2} \)
67 \( 1 - 405.T + 3.00e5T^{2} \)
71 \( 1 - 965.T + 3.57e5T^{2} \)
73 \( 1 - 110.T + 3.89e5T^{2} \)
79 \( 1 - 168.T + 4.93e5T^{2} \)
83 \( 1 - 56.6T + 5.71e5T^{2} \)
89 \( 1 - 574.T + 7.04e5T^{2} \)
97 \( 1 + 966.T + 9.12e5T^{2} \)
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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

−12.36666714462234528763508930089, −11.83290619741898809457073359483, −10.86574481796279466049250595659, −10.09561587741969939131074060747, −8.386991273105921086201568295595, −6.85440708211076355822277827073, −6.39200659422848598377453344074, −4.78263713765196885361548173208, −3.74353362366995200048396450763, −0.45793017319916872744750227499, 0.45793017319916872744750227499, 3.74353362366995200048396450763, 4.78263713765196885361548173208, 6.39200659422848598377453344074, 6.85440708211076355822277827073, 8.386991273105921086201568295595, 10.09561587741969939131074060747, 10.86574481796279466049250595659, 11.83290619741898809457073359483, 12.36666714462234528763508930089

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