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Properties

Label 2-136-8.5-c3-0-22
Degree $2$
Conductor $136$
Sign $-0.603 - 0.797i$
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  + (2.69 + 0.856i)2-s + 8.87i·3-s + (6.53 + 4.61i)4-s + 6.22i·5-s + (−7.59 + 23.9i)6-s + 18.5·7-s + (13.6 + 18.0i)8-s − 51.6·9-s + (−5.33 + 16.7i)10-s − 65.1i·11-s + (−40.9 + 57.9i)12-s − 25.7i·13-s + (49.9 + 15.8i)14-s − 55.2·15-s + (21.3 + 60.3i)16-s + 17·17-s + ⋯
L(s)  = 1  + (0.953 + 0.302i)2-s + 1.70i·3-s + (0.816 + 0.577i)4-s + 0.557i·5-s + (−0.516 + 1.62i)6-s + 0.999·7-s + (0.603 + 0.797i)8-s − 1.91·9-s + (−0.168 + 0.530i)10-s − 1.78i·11-s + (−0.985 + 1.39i)12-s − 0.548i·13-s + (0.952 + 0.302i)14-s − 0.951·15-s + (0.333 + 0.942i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.603 - 0.797i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.603 - 0.797i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.36611 + 2.74728i\)
\(L(\frac12)\) \(\approx\) \(1.36611 + 2.74728i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.69 - 0.856i)T \)
17 \( 1 - 17T \)
good3 \( 1 - 8.87iT - 27T^{2} \)
5 \( 1 - 6.22iT - 125T^{2} \)
7 \( 1 - 18.5T + 343T^{2} \)
11 \( 1 + 65.1iT - 1.33e3T^{2} \)
13 \( 1 + 25.7iT - 2.19e3T^{2} \)
19 \( 1 - 22.0iT - 6.85e3T^{2} \)
23 \( 1 + 34.4T + 1.21e4T^{2} \)
29 \( 1 + 141. iT - 2.43e4T^{2} \)
31 \( 1 + 295.T + 2.97e4T^{2} \)
37 \( 1 - 344. iT - 5.06e4T^{2} \)
41 \( 1 - 282.T + 6.89e4T^{2} \)
43 \( 1 + 183. iT - 7.95e4T^{2} \)
47 \( 1 - 441.T + 1.03e5T^{2} \)
53 \( 1 + 271. iT - 1.48e5T^{2} \)
59 \( 1 - 250. iT - 2.05e5T^{2} \)
61 \( 1 - 160. iT - 2.26e5T^{2} \)
67 \( 1 + 872. iT - 3.00e5T^{2} \)
71 \( 1 + 769.T + 3.57e5T^{2} \)
73 \( 1 + 303.T + 3.89e5T^{2} \)
79 \( 1 + 529.T + 4.93e5T^{2} \)
83 \( 1 - 237. iT - 5.71e5T^{2} \)
89 \( 1 - 1.28e3T + 7.04e5T^{2} \)
97 \( 1 - 1.19e3T + 9.12e5T^{2} \)
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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

−13.49569776683532866024821798991, −11.78482206829157816530386424797, −10.97864522373926721160157878109, −10.49760473737894541948931872637, −8.833777630491666378036445905929, −7.85448789656411607232366602970, −5.98417698296640322164940911126, −5.18834409053648663792361917111, −3.94379635933710425243498359127, −2.99627773871082292278422962995, 1.37850072976835122870082927705, 2.21278746906875050417199519292, 4.47144582283803648011193993115, 5.61623110873153094992864703112, 7.07074787208235067209759749436, 7.54990931196523163890251468537, 9.107978000346931227340495374511, 10.84219422515385141759015579360, 11.87039727681670984846201573136, 12.56773380265737966796948356021

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