↑

Properties

Label 2-136-17.9-c1-0-2
Degree $2$
Conductor $136$
Sign $0.896 - 0.443i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.47 + 1.02i)3-s + (−0.392 + 0.947i)5-s + (−0.893 − 2.15i)7-s + (2.94 + 2.94i)9-s + (−4.81 + 1.99i)11-s − 4.35i·13-s + (−1.94 + 1.94i)15-s + (3.77 − 1.65i)17-s + (−1.89 + 1.89i)19-s − 6.24i·21-s + (3.79 − 1.57i)23-s + (2.79 + 2.79i)25-s + (1.18 + 2.86i)27-s + (−1.46 + 3.54i)29-s + (−10.1 − 4.20i)31-s + ⋯
L(s)  = 1  + (1.42 + 0.591i)3-s + (−0.175 + 0.423i)5-s + (−0.337 − 0.815i)7-s + (0.980 + 0.980i)9-s + (−1.45 + 0.600i)11-s − 1.20i·13-s + (−0.501 + 0.501i)15-s + (0.915 − 0.401i)17-s + (−0.434 + 0.434i)19-s − 1.36i·21-s + (0.791 − 0.327i)23-s + (0.558 + 0.558i)25-s + (0.228 + 0.550i)27-s + (−0.272 + 0.658i)29-s + (−1.82 − 0.754i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43752 + 0.336213i\)
\(L(\frac12)\) \(\approx\) \(1.43752 + 0.336213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.77 + 1.65i)T \)
good3 \( 1 + (-2.47 - 1.02i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.392 - 0.947i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.893 + 2.15i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.81 - 1.99i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 4.35iT - 13T^{2} \)
19 \( 1 + (1.89 - 1.89i)T - 19iT^{2} \)
23 \( 1 + (-3.79 + 1.57i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.46 - 3.54i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (10.1 + 4.20i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.02 - 0.423i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.00 + 2.42i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.86 + 2.86i)T + 43iT^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 + (5.49 - 5.49i)T - 53iT^{2} \)
59 \( 1 + (-7.69 - 7.69i)T + 59iT^{2} \)
61 \( 1 + (-3.04 - 7.35i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 6.44T + 67T^{2} \)
71 \( 1 + (-1.73 - 0.718i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.05 + 12.1i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.31 + 0.957i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.0232 + 0.0232i)T - 83iT^{2} \)
89 \( 1 + 7.34iT - 89T^{2} \)
97 \( 1 + (6.98 - 16.8i)T + (-68.5 - 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.29318560165488093852803221834, −12.74463012301339690684657524069, −10.71118126204176162683304282715, −10.26322000616008347451060440296, −9.183015423356117143738918110667, −7.86328872985431872132486480545, −7.35300673659929424627153858059, −5.23802977968739672414629249235, −3.67664102443138957054644681357, −2.75817612893797966627871000865, 2.19108748764499771531730132277, 3.42046915334637644499662196607, 5.29204815750514299224486327468, 6.89547022056347726969251992230, 8.117956835408423819157067162881, 8.721291433760107039436826320618, 9.668740530203642613701181929959, 11.20009658088354356098968654421, 12.60866983765458545758885516249, 13.05011410809081804477778093624

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