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Properties

Label 2-136-17.9-c1-0-4
Degree $2$
Conductor $136$
Sign $-0.197 + 0.980i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 0.707i)3-s + (0.292 − 0.707i)5-s + (−1.70 − 4.12i)7-s + (0.292 + 0.292i)9-s + (1.70 − 0.707i)11-s + (−1 + 0.999i)15-s + (1 − 4i)17-s + (−4.41 + 4.41i)19-s + 8.24i·21-s + (4.53 − 1.87i)23-s + (3.12 + 3.12i)25-s + (1.82 + 4.41i)27-s + (−0.878 + 2.12i)29-s + (5.12 + 2.12i)31-s − 3.41·33-s + ⋯
L(s)  = 1  + (−0.985 − 0.408i)3-s + (0.130 − 0.316i)5-s + (−0.645 − 1.55i)7-s + (0.0976 + 0.0976i)9-s + (0.514 − 0.213i)11-s + (−0.258 + 0.258i)15-s + (0.242 − 0.970i)17-s + (−1.01 + 1.01i)19-s + 1.79i·21-s + (0.945 − 0.391i)23-s + (0.624 + 0.624i)25-s + (0.351 + 0.849i)27-s + (−0.163 + 0.393i)29-s + (0.919 + 0.381i)31-s − 0.594·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445351 - 0.543824i\)
\(L(\frac12)\) \(\approx\) \(0.445351 - 0.543824i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 4i)T \)
good3 \( 1 + (1.70 + 0.707i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.292 + 0.707i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.70 + 4.12i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.70 + 0.707i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 13T^{2} \)
19 \( 1 + (4.41 - 4.41i)T - 19iT^{2} \)
23 \( 1 + (-4.53 + 1.87i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.878 - 2.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-5.12 - 2.12i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.70 - 0.707i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.70 + 8.94i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.24 + 1.24i)T + 43iT^{2} \)
47 \( 1 + 7.17iT - 47T^{2} \)
53 \( 1 + (-7.82 + 7.82i)T - 53iT^{2} \)
59 \( 1 + (-8.41 - 8.41i)T + 59iT^{2} \)
61 \( 1 + (4.87 + 11.7i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 1.65T + 67T^{2} \)
71 \( 1 + (-2.29 - 0.949i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (5.36 - 12.9i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-12.5 + 5.19i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-1.24 + 1.24i)T - 83iT^{2} \)
89 \( 1 - 9.65iT - 89T^{2} \)
97 \( 1 + (-0.778 + 1.87i)T + (-68.5 - 68.5i)T^{2} \)
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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

−12.87070555189741735501339467034, −11.97065692342899570696881108481, −10.90042371237740653021200544863, −10.09383025157314976992291234179, −8.762924745979933267427006587083, −7.14122774594751221110520638400, −6.53172030298454632147526814749, −5.14431452819139227153559737165, −3.65744991360970000038252171712, −0.844277900644065299776300647648, 2.69085940400383342724202037154, 4.64608879087912381497830631296, 5.91186569801768685588346898984, 6.53538641281901950419130297654, 8.451742630397633008855426735450, 9.449042012786102215074637020806, 10.55329396069843070742757319824, 11.50212147676183983148593346576, 12.30654620011648464313181417029, 13.25100858614872392796178439056

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