Properties

Label 2-136-8.5-c3-0-36
Degree $2$
Conductor $136$
Sign $0.334 + 0.942i$
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  + (2.27 − 1.68i)2-s + 4.93i·3-s + (2.33 − 7.65i)4-s − 13.0i·5-s + (8.29 + 11.2i)6-s + 6.24·7-s + (−7.56 − 21.3i)8-s + 2.68·9-s + (−21.8 − 29.5i)10-s − 46.1i·11-s + (37.7 + 11.5i)12-s + 58.0i·13-s + (14.1 − 10.5i)14-s + 64.1·15-s + (−53.0 − 35.7i)16-s + 17·17-s + ⋯
L(s)  = 1  + (0.803 − 0.594i)2-s + 0.949i·3-s + (0.292 − 0.956i)4-s − 1.16i·5-s + (0.564 + 0.762i)6-s + 0.336·7-s + (−0.334 − 0.942i)8-s + 0.0992·9-s + (−0.691 − 0.934i)10-s − 1.26i·11-s + (0.907 + 0.277i)12-s + 1.23i·13-s + (0.270 − 0.200i)14-s + 1.10·15-s + (−0.829 − 0.558i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.334 + 0.942i$
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: \((2,\ 136,\ (\ :3/2),\ 0.334 + 0.942i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.14780 - 1.51706i\)
\(L(\frac12)\) \(\approx\) \(2.14780 - 1.51706i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.27 + 1.68i)T \)
17 \( 1 - 17T \)
good3 \( 1 - 4.93iT - 27T^{2} \)
5 \( 1 + 13.0iT - 125T^{2} \)
7 \( 1 - 6.24T + 343T^{2} \)
11 \( 1 + 46.1iT - 1.33e3T^{2} \)
13 \( 1 - 58.0iT - 2.19e3T^{2} \)
19 \( 1 + 154. iT - 6.85e3T^{2} \)
23 \( 1 - 200.T + 1.21e4T^{2} \)
29 \( 1 - 224. iT - 2.43e4T^{2} \)
31 \( 1 + 230.T + 2.97e4T^{2} \)
37 \( 1 - 122. iT - 5.06e4T^{2} \)
41 \( 1 + 110.T + 6.89e4T^{2} \)
43 \( 1 - 330. iT - 7.95e4T^{2} \)
47 \( 1 + 100.T + 1.03e5T^{2} \)
53 \( 1 - 477. iT - 1.48e5T^{2} \)
59 \( 1 - 205. iT - 2.05e5T^{2} \)
61 \( 1 + 242. iT - 2.26e5T^{2} \)
67 \( 1 - 30.7iT - 3.00e5T^{2} \)
71 \( 1 - 696.T + 3.57e5T^{2} \)
73 \( 1 - 841.T + 3.89e5T^{2} \)
79 \( 1 - 752.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 + 302.T + 7.04e5T^{2} \)
97 \( 1 + 104.T + 9.12e5T^{2} \)
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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

−12.68186529464575034692653362148, −11.29987361899811384017325783374, −10.91278677930465635227730379830, −9.264535335496346209440639640472, −8.972473574888186744321597341474, −6.82196786586063685267032819051, −5.11630877086192130664887296643, −4.69903575969259158928835219845, −3.30899507964282465366996181909, −1.17688960427253693271996888084, 2.12391811338660214416554118459, 3.60242196421452703572159749580, 5.31176411079516859358615754436, 6.57737664379953200121568077695, 7.37564053152880112263914903213, 8.021729167897592068123816016251, 10.01360508514476306684469174769, 11.14863142282887277276595349703, 12.40706381008447061626460511989, 12.84398518446107975652553595646

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