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Properties

Label 2-136-17.13-c3-0-2
Degree $2$
Conductor $136$
Sign $0.997 - 0.0767i$
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  + (−6.91 − 6.91i)3-s + (8.19 + 8.19i)5-s + (−18.1 + 18.1i)7-s + 68.7i·9-s + (44.7 − 44.7i)11-s + 9.95·13-s − 113. i·15-s + (51.7 − 47.2i)17-s + 108. i·19-s + 250.·21-s + (−4.64 + 4.64i)23-s + 9.47i·25-s + (288. − 288. i)27-s + (140. + 140. i)29-s + (158. + 158. i)31-s + ⋯
L(s)  = 1  + (−1.33 − 1.33i)3-s + (0.733 + 0.733i)5-s + (−0.977 + 0.977i)7-s + 2.54i·9-s + (1.22 − 1.22i)11-s + 0.212·13-s − 1.95i·15-s + (0.738 − 0.674i)17-s + 1.30i·19-s + 2.60·21-s + (−0.0421 + 0.0421i)23-s + 0.0758i·25-s + (2.05 − 2.05i)27-s + (0.900 + 0.900i)29-s + (0.917 + 0.917i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.997 - 0.0767i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.997 - 0.0767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07394 + 0.0412788i\)
\(L(\frac12)\) \(\approx\) \(1.07394 + 0.0412788i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-51.7 + 47.2i)T \)
good3 \( 1 + (6.91 + 6.91i)T + 27iT^{2} \)
5 \( 1 + (-8.19 - 8.19i)T + 125iT^{2} \)
7 \( 1 + (18.1 - 18.1i)T - 343iT^{2} \)
11 \( 1 + (-44.7 + 44.7i)T - 1.33e3iT^{2} \)
13 \( 1 - 9.95T + 2.19e3T^{2} \)
19 \( 1 - 108. iT - 6.85e3T^{2} \)
23 \( 1 + (4.64 - 4.64i)T - 1.21e4iT^{2} \)
29 \( 1 + (-140. - 140. i)T + 2.43e4iT^{2} \)
31 \( 1 + (-158. - 158. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-33.6 - 33.6i)T + 5.06e4iT^{2} \)
41 \( 1 + (55.7 - 55.7i)T - 6.89e4iT^{2} \)
43 \( 1 - 193. iT - 7.95e4T^{2} \)
47 \( 1 - 246.T + 1.03e5T^{2} \)
53 \( 1 - 208. iT - 1.48e5T^{2} \)
59 \( 1 + 163. iT - 2.05e5T^{2} \)
61 \( 1 + (-76.2 + 76.2i)T - 2.26e5iT^{2} \)
67 \( 1 + 292.T + 3.00e5T^{2} \)
71 \( 1 + (-484. - 484. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-90.9 - 90.9i)T + 3.89e5iT^{2} \)
79 \( 1 + (45.3 - 45.3i)T - 4.93e5iT^{2} \)
83 \( 1 - 661. iT - 5.71e5T^{2} \)
89 \( 1 + 657.T + 7.04e5T^{2} \)
97 \( 1 + (-58.0 - 58.0i)T + 9.12e5iT^{2} \)
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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

−12.42161208656750333751525707541, −12.00548450867271715526670547257, −10.94748552014432590649531598298, −9.853393916969453964309068981819, −8.388588939434318298950371152664, −6.79714693533788026423790093478, −6.21933888342609504995098108188, −5.59472495795647220527010478031, −2.91913428715303958771719664183, −1.20480038616695342519747539929, 0.791265901506604944113914760298, 3.89436842702635569243965364320, 4.70067037008604240173020634566, 5.99908069858717600526456422935, 6.83967720377392994028744793604, 9.169573280551749631488568695085, 9.795933134598393230835972938646, 10.40652633931799980799040735630, 11.67918553025865081690496339838, 12.52424389569502529097048078871

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