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Properties

Label 2-136-136.101-c1-0-0
Degree $2$
Conductor $136$
Sign $-0.953 - 0.300i$
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  + (−1.25 + 0.658i)2-s − 0.909·3-s + (1.13 − 1.64i)4-s − 1.54·5-s + (1.13 − 0.598i)6-s + 3.57i·7-s + (−0.334 + 2.80i)8-s − 2.17·9-s + (1.94 − 1.02i)10-s − 5.24·11-s + (−1.03 + 1.49i)12-s + 0.927i·13-s + (−2.35 − 4.47i)14-s + 1.40·15-s + (−1.42 − 3.73i)16-s + (0.764 − 4.05i)17-s + ⋯
L(s)  = 1  + (−0.885 + 0.465i)2-s − 0.524·3-s + (0.566 − 0.823i)4-s − 0.693·5-s + (0.464 − 0.244i)6-s + 1.35i·7-s + (−0.118 + 0.992i)8-s − 0.724·9-s + (0.613 − 0.322i)10-s − 1.58·11-s + (−0.297 + 0.432i)12-s + 0.257i·13-s + (−0.628 − 1.19i)14-s + 0.363·15-s + (−0.357 − 0.933i)16-s + (0.185 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.953 - 0.300i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ -0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0380167 + 0.247304i\)
\(L(\frac12)\) \(\approx\) \(0.0380167 + 0.247304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.658i)T \)
17 \( 1 + (-0.764 + 4.05i)T \)
good3 \( 1 + 0.909T + 3T^{2} \)
5 \( 1 + 1.54T + 5T^{2} \)
7 \( 1 - 3.57iT - 7T^{2} \)
11 \( 1 + 5.24T + 11T^{2} \)
13 \( 1 - 0.927iT - 13T^{2} \)
19 \( 1 - 5.22iT - 19T^{2} \)
23 \( 1 - 5.37iT - 23T^{2} \)
29 \( 1 - 3.00T + 29T^{2} \)
31 \( 1 + 0.336iT - 31T^{2} \)
37 \( 1 - 7.49T + 37T^{2} \)
41 \( 1 + 5.03iT - 41T^{2} \)
43 \( 1 + 8.91iT - 43T^{2} \)
47 \( 1 + 4.93T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 - 4.95iT - 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 1.27iT - 67T^{2} \)
71 \( 1 - 4.52iT - 71T^{2} \)
73 \( 1 - 9.78iT - 73T^{2} \)
79 \( 1 + 7.76iT - 79T^{2} \)
83 \( 1 - 6.80iT - 83T^{2} \)
89 \( 1 + 0.0474T + 89T^{2} \)
97 \( 1 - 9.54iT - 97T^{2} \)
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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

−13.85717333187848830673246067904, −12.16458309066344688768984904436, −11.63975052098211637623301858169, −10.59743979628150837156155395285, −9.398168580011159008420115838538, −8.311941609618965196206616795011, −7.54310204304055476995220850047, −5.89618867618917447022313986282, −5.28199257414150892700440523095, −2.63739890165446744556277217115, 0.33861627654621047591630372432, 2.98942021373007803969971551146, 4.56841815927951491058664018469, 6.44579813713139717878021731462, 7.73434479314339551502808208416, 8.297413728899422973590645612185, 9.981995568718762440114666786516, 10.81753231135354378386940422634, 11.30891934795410015901567029178, 12.61489137576367676672909025043

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