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Properties

Label 2-136-136.11-c3-0-47
Degree $2$
Conductor $136$
Sign $-0.0513 - 0.998i$
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  + (−2.61 − 1.08i)2-s + (−1.54 − 7.75i)3-s + (5.65 + 5.65i)4-s + (−4.36 + 21.9i)6-s + (−8.65 − 20.9i)8-s + (−32.8 + 13.5i)9-s + (−71.4 − 14.2i)11-s + (35.1 − 52.5i)12-s + 64i·16-s + (21.0 + 66.8i)17-s + 100.·18-s + (13.9 + 5.75i)19-s + (171. + 114. i)22-s + (−148. + 99.4i)24-s + (47.8 + 115. i)25-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (−0.296 − 1.49i)3-s + (0.707 + 0.707i)4-s + (−0.296 + 1.49i)6-s + (−0.382 − 0.923i)8-s + (−1.21 + 0.503i)9-s + (−1.95 − 0.389i)11-s + (0.845 − 1.26i)12-s + i·16-s + (0.299 + 0.954i)17-s + 1.31·18-s + (0.167 + 0.0695i)19-s + (1.66 + 1.10i)22-s + (−1.26 + 0.845i)24-s + (0.382 + 0.923i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0513 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0513 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.0513 - 0.998i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.0513 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0307095 + 0.0323306i\)
\(L(\frac12)\) \(\approx\) \(0.0307095 + 0.0323306i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.61 + 1.08i)T \)
17 \( 1 + (-21.0 - 66.8i)T \)
good3 \( 1 + (1.54 + 7.75i)T + (-24.9 + 10.3i)T^{2} \)
5 \( 1 + (-47.8 - 115. i)T^{2} \)
7 \( 1 + (-131. + 316. i)T^{2} \)
11 \( 1 + (71.4 + 14.2i)T + (1.22e3 + 509. i)T^{2} \)
13 \( 1 + 2.19e3iT^{2} \)
19 \( 1 + (-13.9 - 5.75i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (1.12e4 + 4.65e3i)T^{2} \)
29 \( 1 + (9.33e3 + 2.25e4i)T^{2} \)
31 \( 1 + (-2.75e4 + 1.14e4i)T^{2} \)
37 \( 1 + (4.67e4 - 1.93e4i)T^{2} \)
41 \( 1 + (416. - 278. i)T + (2.63e4 - 6.36e4i)T^{2} \)
43 \( 1 + (76.5 - 31.6i)T + (5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 + 1.03e5iT^{2} \)
53 \( 1 + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (251. + 607. i)T + (-1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (8.68e4 - 2.09e5i)T^{2} \)
67 \( 1 - 984. iT - 3.00e5T^{2} \)
71 \( 1 + (3.30e5 - 1.36e5i)T^{2} \)
73 \( 1 + (1.02e3 + 684. i)T + (1.48e5 + 3.59e5i)T^{2} \)
79 \( 1 + (-4.55e5 - 1.88e5i)T^{2} \)
83 \( 1 + (-43.2 + 104. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (1.02e3 + 1.02e3i)T + 7.04e5iT^{2} \)
97 \( 1 + (866. - 1.29e3i)T + (-3.49e5 - 8.43e5i)T^{2} \)
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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

−11.97926690475791669400909423966, −10.99643964392086627710696308030, −10.08081991825998200796148457390, −8.425721270291748733531628362679, −7.82015712225535787725118409838, −6.84636998618020262097592098802, −5.59925490131410356560212308610, −2.96707404707322048067557024538, −1.59509749634747671412828013410, −0.03010470766181190784170711216, 2.78252462022277842042412528404, 4.80816991497069666780295108047, 5.58967859687644983100483879636, 7.26805921562488793584172799163, 8.421703508345492979951170028394, 9.569004689939145383221557122667, 10.31310078661785814969717974126, 10.87112193896159392420879642844, 12.11617440149896054056650609320, 13.76250719463409602805906724060

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