Properties

Label 2-63-63.59-c3-0-20
Degree $2$
Conductor $63$
Sign $-0.764 + 0.644i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.09 + 1.20i)2-s + (−5.12 − 0.866i)3-s + (−1.08 − 1.87i)4-s − 10.7·5-s + (−9.67 − 8.00i)6-s + (−17.6 − 5.71i)7-s − 24.5i·8-s + (25.4 + 8.88i)9-s + (−22.4 − 12.9i)10-s + 26.1i·11-s + (3.91 + 10.5i)12-s + (−29.8 − 17.2i)13-s + (−29.9 − 33.2i)14-s + (54.9 + 9.29i)15-s + (21.0 − 36.4i)16-s + (2.29 − 3.98i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.427i)2-s + (−0.985 − 0.166i)3-s + (−0.135 − 0.233i)4-s − 0.958·5-s + (−0.658 − 0.544i)6-s + (−0.951 − 0.308i)7-s − 1.08i·8-s + (0.944 + 0.328i)9-s + (−0.709 − 0.409i)10-s + 0.716i·11-s + (0.0941 + 0.253i)12-s + (−0.637 − 0.368i)13-s + (−0.572 − 0.634i)14-s + (0.945 + 0.159i)15-s + (0.328 − 0.569i)16-s + (0.0327 − 0.0567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.138044 - 0.378302i\)
\(L(\frac12)\) \(\approx\) \(0.138044 - 0.378302i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.12 + 0.866i)T \)
7 \( 1 + (17.6 + 5.71i)T \)
good2 \( 1 + (-2.09 - 1.20i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 10.7T + 125T^{2} \)
11 \( 1 - 26.1iT - 1.33e3T^{2} \)
13 \( 1 + (29.8 + 17.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-2.29 + 3.98i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-6.74 + 3.89i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 39.1iT - 1.21e4T^{2} \)
29 \( 1 + (-210. + 121. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (269. - 155. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (28.3 + 49.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-148. + 256. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (60.7 + 105. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-112. + 195. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (307. + 177. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (135. + 234. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (649. + 374. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (497. + 861. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.13e3iT - 3.57e5T^{2} \)
73 \( 1 + (-1.07e3 - 620. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (216. - 375. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-492. - 853. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-75.6 - 131. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-781. + 451. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.01843226132826181532396529147, −12.69970816625376226446911035253, −12.24333561481764750574907428944, −10.65975596346793226834258961870, −9.665821360850711536719450454678, −7.44071833434579132282085561897, −6.54983420200208698367429226878, −5.14680294479637744330574805158, −3.94760775755522230600696885233, −0.24278761807542247000336413102, 3.32484089649725963572300961770, 4.56931519197361377785030865268, 6.00573609526784001050652685835, 7.58736840499040455927275643922, 9.228133890220511843395121986286, 10.81661868552168745049577150483, 11.82780849343642038221712286549, 12.40244376859641180974304407377, 13.43915563174455887375821777147, 14.90667182294162340203951062168

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