↑

Properties

Label 2-63-63.25-c3-0-3
Degree $2$
Conductor $63$
Sign $-0.584 - 0.811i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.590·2-s + (4.35 + 2.83i)3-s − 7.65·4-s + (−5.49 + 9.51i)5-s + (−2.57 − 1.67i)6-s + (−16.7 + 7.98i)7-s + 9.24·8-s + (10.9 + 24.6i)9-s + (3.24 − 5.61i)10-s + (−22.8 − 39.6i)11-s + (−33.3 − 21.6i)12-s + (37.1 + 64.4i)13-s + (9.87 − 4.71i)14-s + (−50.8 + 25.8i)15-s + 55.7·16-s + (−40.4 + 70.0i)17-s + ⋯
L(s)  = 1  − 0.208·2-s + (0.837 + 0.545i)3-s − 0.956·4-s + (−0.491 + 0.850i)5-s + (−0.175 − 0.113i)6-s + (−0.902 + 0.431i)7-s + 0.408·8-s + (0.404 + 0.914i)9-s + (0.102 − 0.177i)10-s + (−0.627 − 1.08i)11-s + (−0.801 − 0.521i)12-s + (0.793 + 1.37i)13-s + (0.188 − 0.0900i)14-s + (−0.875 + 0.444i)15-s + 0.871·16-s + (−0.577 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.584 - 0.811i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.432462 + 0.844330i\)
\(L(\frac12)\) \(\approx\) \(0.432462 + 0.844330i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.35 - 2.83i)T \)
7 \( 1 + (16.7 - 7.98i)T \)
good2 \( 1 + 0.590T + 8T^{2} \)
5 \( 1 + (5.49 - 9.51i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (22.8 + 39.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-37.1 - 64.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (40.4 - 70.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-8.05 - 13.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-67.0 + 116. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-114. + 198. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 91.6T + 2.97e4T^{2} \)
37 \( 1 + (-5.76 - 9.98i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-56.3 - 97.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (248. - 430. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 82.9T + 1.03e5T^{2} \)
53 \( 1 + (247. - 427. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 259.T + 2.05e5T^{2} \)
61 \( 1 - 161.T + 2.26e5T^{2} \)
67 \( 1 + 293.T + 3.00e5T^{2} \)
71 \( 1 - 249.T + 3.57e5T^{2} \)
73 \( 1 + (398. - 691. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 740.T + 4.93e5T^{2} \)
83 \( 1 + (-215. + 373. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (224. + 388. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-125. + 218. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−14.80424736218206431824948257945, −13.79657591386681031301395471463, −13.02748109105774960992533312941, −11.16399489497785489757562890101, −10.11034598967797527672396950245, −8.935687863773689485734570690237, −8.181240485932235841499130004950, −6.35854756308325386442699684359, −4.27068991647862750730507050199, −3.06889058489528832935287205797, 0.67406694342833963804710943749, 3.40358163849599468876050433498, 4.95674762331370174047793913296, 7.16435025209155190666235276287, 8.258970107916059846346492887647, 9.203314434407237113906441613869, 10.24079752061046575397577520497, 12.38872945442684137055684093618, 13.07598200692772771100360718113, 13.68553488639702346144692928129

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