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Properties

Label 2-136-8.5-c1-0-6
Degree $2$
Conductor $136$
Sign $0.911 + 0.411i$
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  + (−1.40 − 0.199i)2-s − 1.72i·3-s + (1.92 + 0.557i)4-s + 3.30i·5-s + (−0.342 + 2.40i)6-s + 3.45·7-s + (−2.57 − 1.16i)8-s + 0.0412·9-s + (0.657 − 4.62i)10-s − 3.09i·11-s + (0.958 − 3.30i)12-s − 2.50i·13-s + (−4.84 − 0.688i)14-s + 5.68·15-s + (3.37 + 2.14i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.990 − 0.140i)2-s − 0.993i·3-s + (0.960 + 0.278i)4-s + 1.47i·5-s + (−0.139 + 0.983i)6-s + 1.30·7-s + (−0.911 − 0.411i)8-s + 0.0137·9-s + (0.207 − 1.46i)10-s − 0.933i·11-s + (0.276 − 0.953i)12-s − 0.695i·13-s + (−1.29 − 0.183i)14-s + 1.46·15-s + (0.844 + 0.535i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807568 - 0.173665i\)
\(L(\frac12)\) \(\approx\) \(0.807568 - 0.173665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.199i)T \)
17 \( 1 + T \)
good3 \( 1 + 1.72iT - 3T^{2} \)
5 \( 1 - 3.30iT - 5T^{2} \)
7 \( 1 - 3.45T + 7T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 + 2.50iT - 13T^{2} \)
19 \( 1 - 4.43iT - 19T^{2} \)
23 \( 1 - 4.14T + 23T^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 6.74iT - 37T^{2} \)
41 \( 1 + 6.99T + 41T^{2} \)
43 \( 1 - 2.22iT - 43T^{2} \)
47 \( 1 + 3.31T + 47T^{2} \)
53 \( 1 - 6.19iT - 53T^{2} \)
59 \( 1 + 8.83iT - 59T^{2} \)
61 \( 1 - 14.3iT - 61T^{2} \)
67 \( 1 + 6.44iT - 67T^{2} \)
71 \( 1 + 5.66T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + 8.14T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 - 8.36T + 97T^{2} \)
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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

−13.01363207208540551430688427167, −11.81155414291056886793185186900, −10.95489121097174456351642700945, −10.41122035477164107766173898306, −8.692112019202758616395295041614, −7.68531484241958642187272405952, −7.08289442999433584116826249080, −5.88250195046559482970596086136, −3.17448292594507152929346998058, −1.66874057138773341688402114835, 1.67006832066013342948335199098, 4.48771634430499630918129281431, 5.14276881875489496197394442116, 7.12796203523267026194632211472, 8.379563772110845690292534983386, 9.123049725143780302706655389364, 9.902106000773040523550947952828, 11.13274644609344110332210541812, 11.89797820999985082053957555255, 13.17959968952411165194405350199

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