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Properties

Label 2-136-17.15-c1-0-0
Degree $2$
Conductor $136$
Sign $0.0465 - 0.998i$
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 + 2.41i)3-s + (−2.70 + 1.12i)5-s + (1 + 0.414i)7-s + (−2.70 + 2.70i)9-s + (1.58 − 3.82i)11-s + 2.58i·13-s + (−5.41 − 5.41i)15-s + (2.82 − 3i)17-s + (5.41 + 5.41i)19-s + 2.82i·21-s + (2.41 − 5.82i)23-s + (2.53 − 2.53i)25-s + (−2 − 0.828i)27-s + (−5.12 + 2.12i)29-s + (−0.414 − i)31-s + ⋯
L(s)  = 1  + (0.577 + 1.39i)3-s + (−1.21 + 0.501i)5-s + (0.377 + 0.156i)7-s + (−0.902 + 0.902i)9-s + (0.478 − 1.15i)11-s + 0.717i·13-s + (−1.39 − 1.39i)15-s + (0.685 − 0.727i)17-s + (1.24 + 1.24i)19-s + 0.617i·21-s + (0.503 − 1.21i)23-s + (0.507 − 0.507i)25-s + (−0.384 − 0.159i)27-s + (−0.951 + 0.393i)29-s + (−0.0743 − 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.0465 - 0.998i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.0465 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820529 + 0.783211i\)
\(L(\frac12)\) \(\approx\) \(0.820529 + 0.783211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-2.82 + 3i)T \)
good3 \( 1 + (-1 - 2.41i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (2.70 - 1.12i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1 - 0.414i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.58 + 3.82i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 2.58iT - 13T^{2} \)
19 \( 1 + (-5.41 - 5.41i)T + 19iT^{2} \)
23 \( 1 + (-2.41 + 5.82i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (5.12 - 2.12i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.414 + i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (4.12 + 9.94i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (8.53 + 3.53i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.242 + 0.242i)T - 43iT^{2} \)
47 \( 1 + 1.17iT - 47T^{2} \)
53 \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \)
59 \( 1 + (5.07 - 5.07i)T - 59iT^{2} \)
61 \( 1 + (-4.12 - 1.70i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 2.82T + 67T^{2} \)
71 \( 1 + (-3.48 - 8.41i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.36 + 3.05i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.58 + 3.82i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (9.07 + 9.07i)T + 83iT^{2} \)
89 \( 1 - 4.24iT - 89T^{2} \)
97 \( 1 + (4.12 - 1.70i)T + (68.5 - 68.5i)T^{2} \)
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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

−14.03501867750873375753447977163, −12.06398573920807827975449572597, −11.33183586978126775793431701318, −10.44304698906596204823128437738, −9.242893830145144197679802043665, −8.406293465723830743600754873592, −7.25455182193079029685852750048, −5.39445222000120628302248431406, −3.96574153060757612134294273856, −3.28389998045347009998040884321, 1.38053787386229656749886275517, 3.38106650265952903260148178650, 5.01373469425691523005371018032, 6.92950380634399347759571040245, 7.66076909275871127418207004193, 8.318270024138588281427088406208, 9.647502514358346853236446098037, 11.41418866031770305669872466127, 12.09817927172330157850610281754, 12.90901732034320426636070326305

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