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Properties

Label 2-640-16.13-c3-0-17
Degree $2$
Conductor $640$
Sign $0.262 + 0.964i$
Analytic cond. $37.7612$
Root an. cond. $6.14501$
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  + (−4.63 − 4.63i)3-s + (−3.53 + 3.53i)5-s + 8.38i·7-s + 15.9i·9-s + (−32.6 + 32.6i)11-s + (−16.0 − 16.0i)13-s + 32.7·15-s + 31.3·17-s + (−81.4 − 81.4i)19-s + (38.8 − 38.8i)21-s + 183. i·23-s − 25.0i·25-s + (−51.3 + 51.3i)27-s + (138. + 138. i)29-s + 168.·31-s + ⋯
L(s)  = 1  + (−0.891 − 0.891i)3-s + (−0.316 + 0.316i)5-s + 0.452i·7-s + 0.589i·9-s + (−0.894 + 0.894i)11-s + (−0.342 − 0.342i)13-s + 0.563·15-s + 0.446·17-s + (−0.983 − 0.983i)19-s + (0.403 − 0.403i)21-s + 1.66i·23-s − 0.200i·25-s + (−0.366 + 0.366i)27-s + (0.886 + 0.886i)29-s + 0.976·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.262 + 0.964i$
Analytic conductor: \(37.7612\)
Root analytic conductor: \(6.14501\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :3/2),\ 0.262 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7948200973\)
\(L(\frac12)\) \(\approx\) \(0.7948200973\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (3.53 - 3.53i)T \)
good3 \( 1 + (4.63 + 4.63i)T + 27iT^{2} \)
7 \( 1 - 8.38iT - 343T^{2} \)
11 \( 1 + (32.6 - 32.6i)T - 1.33e3iT^{2} \)
13 \( 1 + (16.0 + 16.0i)T + 2.19e3iT^{2} \)
17 \( 1 - 31.3T + 4.91e3T^{2} \)
19 \( 1 + (81.4 + 81.4i)T + 6.85e3iT^{2} \)
23 \( 1 - 183. iT - 1.21e4T^{2} \)
29 \( 1 + (-138. - 138. i)T + 2.43e4iT^{2} \)
31 \( 1 - 168.T + 2.97e4T^{2} \)
37 \( 1 + (-154. + 154. i)T - 5.06e4iT^{2} \)
41 \( 1 + 75.7iT - 6.89e4T^{2} \)
43 \( 1 + (-145. + 145. i)T - 7.95e4iT^{2} \)
47 \( 1 + 530.T + 1.03e5T^{2} \)
53 \( 1 + (-9.51 + 9.51i)T - 1.48e5iT^{2} \)
59 \( 1 + (-87.5 + 87.5i)T - 2.05e5iT^{2} \)
61 \( 1 + (170. + 170. i)T + 2.26e5iT^{2} \)
67 \( 1 + (480. + 480. i)T + 3.00e5iT^{2} \)
71 \( 1 + 630. iT - 3.57e5T^{2} \)
73 \( 1 + 546. iT - 3.89e5T^{2} \)
79 \( 1 - 496.T + 4.93e5T^{2} \)
83 \( 1 + (205. + 205. i)T + 5.71e5iT^{2} \)
89 \( 1 + 779. iT - 7.04e5T^{2} \)
97 \( 1 + 1.27e3T + 9.12e5T^{2} \)
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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

−10.17897904892779621110124003958, −9.141940526696037019253634314601, −7.908418803318216005531288314315, −7.29196276163467863212174736140, −6.47414445138909046771163160349, −5.51963486731715467260213001435, −4.68329935454653205161420334971, −3.06209891156302336398081856074, −1.88585974587993947046992682902, −0.39535866499737361316959766281, 0.72835808029446332310125577924, 2.70103548339626909412668356191, 4.15965664719047037917014847569, 4.67479624387867557196803786320, 5.75575421934032665036858909798, 6.51543869295383522688083584907, 7.970959315864153278708871100910, 8.456856548342554915312838191624, 9.917308309967075978166527976542, 10.31276594085016263198300795201

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