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Properties

Label 2-136-8.5-c1-0-10
Degree $2$
Conductor $136$
Sign $0.953 + 0.301i$
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  + (0.578 + 1.29i)2-s − 2.34i·3-s + (−1.33 + 1.49i)4-s − 3.12i·5-s + (3.02 − 1.35i)6-s + 2.86·7-s + (−2.69 − 0.852i)8-s − 2.50·9-s + (4.02 − 1.80i)10-s + 5.06i·11-s + (3.50 + 3.12i)12-s − 2.04i·13-s + (1.66 + 3.70i)14-s − 7.32·15-s + (−0.460 − 3.97i)16-s − 17-s + ⋯
L(s)  = 1  + (0.409 + 0.912i)2-s − 1.35i·3-s + (−0.665 + 0.746i)4-s − 1.39i·5-s + (1.23 − 0.554i)6-s + 1.08·7-s + (−0.953 − 0.301i)8-s − 0.834·9-s + (1.27 − 0.571i)10-s + 1.52i·11-s + (1.01 + 0.900i)12-s − 0.566i·13-s + (0.443 + 0.989i)14-s − 1.89·15-s + (−0.115 − 0.993i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.953 + 0.301i$
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: \((2,\ 136,\ (\ :1/2),\ 0.953 + 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28858 - 0.198824i\)
\(L(\frac12)\) \(\approx\) \(1.28858 - 0.198824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.578 - 1.29i)T \)
17 \( 1 + T \)
good3 \( 1 + 2.34iT - 3T^{2} \)
5 \( 1 + 3.12iT - 5T^{2} \)
7 \( 1 - 2.86T + 7T^{2} \)
11 \( 1 - 5.06iT - 11T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
19 \( 1 - 6.32iT - 19T^{2} \)
23 \( 1 + 3.18T + 23T^{2} \)
29 \( 1 - 5.45iT - 29T^{2} \)
31 \( 1 - 6.19T + 31T^{2} \)
37 \( 1 + 1.57iT - 37T^{2} \)
41 \( 1 + 0.733T + 41T^{2} \)
43 \( 1 - 5.97iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 - 0.268iT - 59T^{2} \)
61 \( 1 - 2.58iT - 61T^{2} \)
67 \( 1 - 0.116iT - 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 0.815T + 79T^{2} \)
83 \( 1 + 3.81iT - 83T^{2} \)
89 \( 1 + 9.87T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−12.99380968319083843657626265157, −12.51105717064060389729313957329, −11.86941538240544527282822256147, −9.735381674535354979284570940986, −8.171612563845949426139380117015, −8.030988153150244832294300108190, −6.75200001306946369230536700016, −5.37608417903037233516422793347, −4.44993439731418928622557456988, −1.62663863588844996542326581052, 2.71501104309801672827002861464, 3.89762786351718955548572029576, 5.00129676627912791045025483800, 6.38223923768757146973586558102, 8.364836417026384168258953844931, 9.504196298153077097887305075724, 10.54859027884149800243048073237, 11.15447002362822804529017184378, 11.60314924277487063234613426176, 13.67018896286010135819998973333

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