↑

Properties

Label 2-63-63.58-c3-0-1
Degree $2$
Conductor $63$
Sign $-0.455 - 0.890i$
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  − 2.65·2-s + (−0.794 − 5.13i)3-s − 0.952·4-s + (−3.67 − 6.37i)5-s + (2.10 + 13.6i)6-s + (−1.32 + 18.4i)7-s + 23.7·8-s + (−25.7 + 8.15i)9-s + (9.76 + 16.9i)10-s + (−5.67 + 9.82i)11-s + (0.756 + 4.88i)12-s + (−30.2 + 52.4i)13-s + (3.51 − 49.0i)14-s + (−29.7 + 23.9i)15-s − 55.4·16-s + (−5.29 − 9.16i)17-s + ⋯
L(s)  = 1  − 0.938·2-s + (−0.152 − 0.988i)3-s − 0.119·4-s + (−0.328 − 0.569i)5-s + (0.143 + 0.927i)6-s + (−0.0715 + 0.997i)7-s + 1.05·8-s + (−0.953 + 0.302i)9-s + (0.308 + 0.534i)10-s + (−0.155 + 0.269i)11-s + (0.0181 + 0.117i)12-s + (−0.646 + 1.11i)13-s + (0.0671 − 0.936i)14-s + (−0.512 + 0.412i)15-s − 0.866·16-s + (−0.0755 − 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.455 - 0.890i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.455 - 0.890i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0704252 + 0.115182i\)
\(L(\frac12)\) \(\approx\) \(0.0704252 + 0.115182i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.794 + 5.13i)T \)
7 \( 1 + (1.32 - 18.4i)T \)
good2 \( 1 + 2.65T + 8T^{2} \)
5 \( 1 + (3.67 + 6.37i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (5.67 - 9.82i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (30.2 - 52.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (5.29 + 9.16i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (41.6 - 72.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (43.8 + 75.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (67.6 + 117. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 53.5T + 2.97e4T^{2} \)
37 \( 1 + (149. - 259. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (194. - 336. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (215. + 373. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 511.T + 1.03e5T^{2} \)
53 \( 1 + (-116. - 201. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 62.3T + 2.05e5T^{2} \)
61 \( 1 - 340.T + 2.26e5T^{2} \)
67 \( 1 + 549.T + 3.00e5T^{2} \)
71 \( 1 - 505.T + 3.57e5T^{2} \)
73 \( 1 + (99.1 + 171. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + (722. + 1.25e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (507. - 878. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-551. - 954. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−14.77989268815595541641801659618, −13.55612393173418304028954260379, −12.40647468040592608674038797663, −11.67803382807522222870531965978, −9.970852629935266273852199438466, −8.721351901729081784933525915683, −8.055150643072566997602020343188, −6.58498628442508522896080208765, −4.83425987580647469720200430443, −1.91629088511924473651048150783, 0.12674495834253038813901687744, 3.53652445186285207904370083939, 5.03942723785817799346921673373, 7.14571531636443935693624262732, 8.338908522110279004418546346908, 9.652510596264191953308633908996, 10.50558877362443881603038181990, 11.16869735545934666176351169166, 13.09885100184628678226152445364, 14.32570041564434035576505115822

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