↑

Properties

Label 2-102-51.26-c4-0-8
Degree $2$
Conductor $102$
Sign $-0.169 - 0.985i$
Analytic cond. $10.5437$
Root an. cond. $3.24711$
Motivic weight $4$
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 + 2i)2-s + (−8.98 + 0.470i)3-s + 8i·4-s + (44.5 + 18.4i)5-s + (−18.9 − 17.0i)6-s + (−18.9 − 45.7i)7-s + (−16 + 16i)8-s + (80.5 − 8.46i)9-s + (52.2 + 126. i)10-s + (52.1 + 125. i)11-s + (−3.76 − 71.9i)12-s + 104. i·13-s + (53.6 − 129. i)14-s + (−409. − 144. i)15-s − 64·16-s + (−189. + 218. i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.998 + 0.0523i)3-s + 0.5i·4-s + (1.78 + 0.738i)5-s + (−0.525 − 0.473i)6-s + (−0.386 − 0.934i)7-s + (−0.250 + 0.250i)8-s + (0.994 − 0.104i)9-s + (0.522 + 1.26i)10-s + (0.430 + 1.04i)11-s + (−0.0261 − 0.499i)12-s + 0.621i·13-s + (0.273 − 0.660i)14-s + (−1.81 − 0.644i)15-s − 0.250·16-s + (−0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(10.5437\)
Root analytic conductor: \(3.24711\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :2),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.30851 + 1.55278i\)
\(L(\frac12)\) \(\approx\) \(1.30851 + 1.55278i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
3 \( 1 + (8.98 - 0.470i)T \)
17 \( 1 + (189. - 218. i)T \)
good5 \( 1 + (-44.5 - 18.4i)T + (441. + 441. i)T^{2} \)
7 \( 1 + (18.9 + 45.7i)T + (-1.69e3 + 1.69e3i)T^{2} \)
11 \( 1 + (-52.1 - 125. i)T + (-1.03e4 + 1.03e4i)T^{2} \)
13 \( 1 - 104. iT - 2.85e4T^{2} \)
19 \( 1 + (61.0 - 61.0i)T - 1.30e5iT^{2} \)
23 \( 1 + (-130. - 314. i)T + (-1.97e5 + 1.97e5i)T^{2} \)
29 \( 1 + (364. + 151. i)T + (5.00e5 + 5.00e5i)T^{2} \)
31 \( 1 + (-108. - 44.9i)T + (6.53e5 + 6.53e5i)T^{2} \)
37 \( 1 + (-685. - 284. i)T + (1.32e6 + 1.32e6i)T^{2} \)
41 \( 1 + (-2.73e3 + 1.13e3i)T + (1.99e6 - 1.99e6i)T^{2} \)
43 \( 1 + (2.21e3 + 2.21e3i)T + 3.41e6iT^{2} \)
47 \( 1 + 798.T + 4.87e6T^{2} \)
53 \( 1 + (-1.47e3 - 1.47e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-3.40e3 + 3.40e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (1.82e3 + 4.39e3i)T + (-9.79e6 + 9.79e6i)T^{2} \)
67 \( 1 - 4.36e3T + 2.01e7T^{2} \)
71 \( 1 + (378. - 913. i)T + (-1.79e7 - 1.79e7i)T^{2} \)
73 \( 1 + (3.87e3 - 9.36e3i)T + (-2.00e7 - 2.00e7i)T^{2} \)
79 \( 1 + (-2.33e3 + 966. i)T + (2.75e7 - 2.75e7i)T^{2} \)
83 \( 1 + (-280. - 280. i)T + 4.74e7iT^{2} \)
89 \( 1 + 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + (1.30e3 - 3.14e3i)T + (-6.25e7 - 6.25e7i)T^{2} \)
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

−13.40210869472643080584003413918, −12.68781992400602686792402903077, −11.18427668082177111727714835582, −10.20148266706987408787156891830, −9.447148411354188147125844294776, −7.04170537398499561198670091342, −6.58875955674551063009359094937, −5.51624291177110192658070351074, −4.09930413468744466489246408955, −1.85226494448179635231560411864, 0.944504611808127347214581376584, 2.51627307827580254762998641786, 4.85093775269648298584946579289, 5.79399154242706962353758903178, 6.36127495492041783671327431932, 8.900737814433780414596240030585, 9.694111565862174945936040711064, 10.79413705404830384582662066845, 11.88386125815956757142809943763, 12.99873338765868067319181441254

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