Properties

Label 2-63-63.41-c3-0-15
Degree $2$
Conductor $63$
Sign $-0.895 + 0.445i$
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  + (−4.67 + 2.70i)2-s + (2.75 − 4.40i)3-s + (10.5 − 18.3i)4-s + (−3.43 + 5.95i)5-s + (−1.00 + 28.0i)6-s + (−17.3 + 6.49i)7-s + 71.1i·8-s + (−11.8 − 24.2i)9-s − 37.1i·10-s + (−10.9 + 6.30i)11-s + (−51.5 − 97.2i)12-s + (−22.5 − 13.0i)13-s + (63.5 − 77.2i)14-s + (16.7 + 31.5i)15-s + (−107. − 186. i)16-s − 124.·17-s + ⋯
L(s)  = 1  + (−1.65 + 0.954i)2-s + (0.530 − 0.847i)3-s + (1.32 − 2.29i)4-s + (−0.307 + 0.532i)5-s + (−0.0680 + 1.90i)6-s + (−0.936 + 0.350i)7-s + 3.14i·8-s + (−0.437 − 0.899i)9-s − 1.17i·10-s + (−0.299 + 0.172i)11-s + (−1.24 − 2.33i)12-s + (−0.481 − 0.277i)13-s + (1.21 − 1.47i)14-s + (0.288 + 0.543i)15-s + (−1.68 − 2.91i)16-s − 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.895 + 0.445i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.895 + 0.445i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00464086 - 0.0197543i\)
\(L(\frac12)\) \(\approx\) \(0.00464086 - 0.0197543i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.75 + 4.40i)T \)
7 \( 1 + (17.3 - 6.49i)T \)
good2 \( 1 + (4.67 - 2.70i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (3.43 - 5.95i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (10.9 - 6.30i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (22.5 + 13.0i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 124.T + 4.91e3T^{2} \)
19 \( 1 - 41.3iT - 6.85e3T^{2} \)
23 \( 1 + (97.6 + 56.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-114. + 66.3i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (155. + 89.5i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 148.T + 5.06e4T^{2} \)
41 \( 1 + (93.6 - 162. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-99.1 - 171. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-92.1 - 159. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 359. iT - 1.48e5T^{2} \)
59 \( 1 + (-182. + 315. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-300. + 173. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (182. - 315. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 565. iT - 3.57e5T^{2} \)
73 \( 1 - 737. iT - 3.89e5T^{2} \)
79 \( 1 + (451. + 782. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (382. + 662. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 395.T + 7.04e5T^{2} \)
97 \( 1 + (-243. + 140. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.50605065422730675544151800872, −12.97889266164736971660934437320, −11.40512655668791905281766102354, −10.05105395770353641284713003339, −9.037798753516661471048038449914, −7.981953188353379976754907384293, −6.95454410151386757436675259464, −6.14693359573326964028631698988, −2.38239216458332558963287310910, −0.02024898086410944749359484021, 2.55874626847377763625797773316, 4.08069182292693136523859557114, 7.11407805940947280732072785891, 8.520262752921377355643448237432, 9.194246107683991261715086122550, 10.19744575265368768131390910579, 11.06391062824174314816201724183, 12.35251348688042807489012803883, 13.52035460222902345109813186214, 15.62225991199584265930578859494

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