Properties

Label 2-63-63.58-c3-0-14
Degree $2$
Conductor $63$
Sign $0.911 - 0.411i$
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  + 5.43·2-s + (−0.0704 + 5.19i)3-s + 21.4·4-s + (−6.55 − 11.3i)5-s + (−0.382 + 28.2i)6-s + (−18.0 + 3.92i)7-s + 73.2·8-s + (−26.9 − 0.732i)9-s + (−35.6 − 61.6i)10-s + (9.12 − 15.8i)11-s + (−1.51 + 111. i)12-s + (−12.9 + 22.3i)13-s + (−98.2 + 21.3i)14-s + (59.4 − 33.2i)15-s + 225.·16-s + (1.04 + 1.80i)17-s + ⋯
L(s)  = 1  + 1.91·2-s + (−0.0135 + 0.999i)3-s + 2.68·4-s + (−0.586 − 1.01i)5-s + (−0.0260 + 1.91i)6-s + (−0.977 + 0.212i)7-s + 3.23·8-s + (−0.999 − 0.0271i)9-s + (−1.12 − 1.95i)10-s + (0.250 − 0.433i)11-s + (−0.0364 + 2.68i)12-s + (−0.275 + 0.477i)13-s + (−1.87 + 0.407i)14-s + (1.02 − 0.572i)15-s + 3.52·16-s + (0.0148 + 0.0257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.911 - 0.411i$
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: \((2,\ 63,\ (\ :3/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.40266 + 0.733358i\)
\(L(\frac12)\) \(\approx\) \(3.40266 + 0.733358i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0704 - 5.19i)T \)
7 \( 1 + (18.0 - 3.92i)T \)
good2 \( 1 - 5.43T + 8T^{2} \)
5 \( 1 + (6.55 + 11.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-9.12 + 15.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (12.9 - 22.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-1.04 - 1.80i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.2 + 17.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (34.2 + 59.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-132. - 229. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 30.4T + 2.97e4T^{2} \)
37 \( 1 + (143. - 249. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (152. - 263. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (119. + 207. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 419.T + 1.03e5T^{2} \)
53 \( 1 + (114. + 198. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 799.T + 2.05e5T^{2} \)
61 \( 1 - 94.2T + 2.26e5T^{2} \)
67 \( 1 + 569.T + 3.00e5T^{2} \)
71 \( 1 + 211.T + 3.57e5T^{2} \)
73 \( 1 + (314. + 544. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 334.T + 4.93e5T^{2} \)
83 \( 1 + (-163. - 282. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-160. + 278. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (60.9 + 105. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−14.52169542441629914250061782846, −13.46289438404510129307984976311, −12.33903287062436094059591407076, −11.69893403499934184180953634974, −10.32382606991116591119850472277, −8.676370920811725013888622180522, −6.64752972766917341188872515926, −5.30612779440055309372976748184, −4.30306097997379285568534473133, −3.15108566171468558746901612597, 2.59678930602122138109344205852, 3.75362066562602384648727834433, 5.78612869596757785666631135755, 6.83856638949810253716637316323, 7.52705494674171075802180021491, 10.41591888059076192998208959252, 11.63834806778788457887507690039, 12.35499071874276989514798044348, 13.32802587114808910219236149609, 14.17293106986725052031780525806

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