Properties

Label 2-136-17.16-c3-0-4
Degree $2$
Conductor $136$
Sign $-0.988 - 0.153i$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.52i·3-s + 18.2i·5-s + 13.8i·7-s − 45.6·9-s − 60.8i·11-s + 61.8·13-s − 155.·15-s + (69.2 + 10.7i)17-s − 40.0·19-s − 118.·21-s − 4.88i·23-s − 208.·25-s − 158. i·27-s + 113. i·29-s + 95.1i·31-s + ⋯
L(s)  = 1  + 1.64i·3-s + 1.63i·5-s + 0.749i·7-s − 1.69·9-s − 1.66i·11-s + 1.31·13-s − 2.68·15-s + (0.988 + 0.153i)17-s − 0.483·19-s − 1.22·21-s − 0.0443i·23-s − 1.67·25-s − 1.13i·27-s + 0.724i·29-s + 0.551i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.988 - 0.153i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.988 - 0.153i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.117044 + 1.51860i\)
\(L(\frac12)\) \(\approx\) \(0.117044 + 1.51860i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-69.2 - 10.7i)T \)
good3 \( 1 - 8.52iT - 27T^{2} \)
5 \( 1 - 18.2iT - 125T^{2} \)
7 \( 1 - 13.8iT - 343T^{2} \)
11 \( 1 + 60.8iT - 1.33e3T^{2} \)
13 \( 1 - 61.8T + 2.19e3T^{2} \)
19 \( 1 + 40.0T + 6.85e3T^{2} \)
23 \( 1 + 4.88iT - 1.21e4T^{2} \)
29 \( 1 - 113. iT - 2.43e4T^{2} \)
31 \( 1 - 95.1iT - 2.97e4T^{2} \)
37 \( 1 + 273. iT - 5.06e4T^{2} \)
41 \( 1 - 446. iT - 6.89e4T^{2} \)
43 \( 1 + 274.T + 7.95e4T^{2} \)
47 \( 1 + 27.6T + 1.03e5T^{2} \)
53 \( 1 - 488.T + 1.48e5T^{2} \)
59 \( 1 - 266.T + 2.05e5T^{2} \)
61 \( 1 + 502. iT - 2.26e5T^{2} \)
67 \( 1 + 1.00e3T + 3.00e5T^{2} \)
71 \( 1 + 724. iT - 3.57e5T^{2} \)
73 \( 1 - 188. iT - 3.89e5T^{2} \)
79 \( 1 - 48.3iT - 4.93e5T^{2} \)
83 \( 1 - 1.38e3T + 5.71e5T^{2} \)
89 \( 1 + 50.3T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3iT - 9.12e5T^{2} \)
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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

−13.59425980682325903668440097420, −11.68846355760174908505275090162, −10.83829323919932253192927436805, −10.48696179994705902328067038344, −9.175086423898327236045514886841, −8.252548746551240338377505165413, −6.34142348563780710967914527268, −5.54583112191308759798444194813, −3.64993557346630131539833307498, −3.06683052167864925356585821919, 0.823349142759307950569335209607, 1.77445614502147435550451406196, 4.24030677149390960962996021225, 5.66883136126271393592288773834, 6.99226067915444419689384097529, 7.896968148191090168408693384060, 8.769241921716423703232518565157, 10.10825337644294320039660460715, 11.81968792946952053061492536457, 12.36791717669334295772223052056

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