Properties

Degree $2$
Conductor $63$
Sign $0.958 + 0.283i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.62 − 4.54i)2-s + (−4.41 + 2.73i)3-s + (−9.74 + 16.8i)4-s + (2.03 − 3.51i)5-s + (24.0 + 12.8i)6-s + (3.5 + 6.06i)7-s + 60.2·8-s + (12.0 − 24.1i)9-s − 21.2·10-s + (7.57 + 13.1i)11-s + (−3.11 − 101. i)12-s + (−24.2 + 42.0i)13-s + (18.3 − 31.7i)14-s + (0.648 + 21.0i)15-s + (−80.0 − 138. i)16-s + 107.·17-s + ⋯
L(s)  = 1  + (−0.926 − 1.60i)2-s + (−0.850 + 0.526i)3-s + (−1.21 + 2.11i)4-s + (0.181 − 0.314i)5-s + (1.63 + 0.877i)6-s + (0.188 + 0.327i)7-s + 2.66·8-s + (0.445 − 0.895i)9-s − 0.673·10-s + (0.207 + 0.359i)11-s + (−0.0749 − 2.43i)12-s + (−0.518 + 0.897i)13-s + (0.350 − 0.606i)14-s + (0.0111 + 0.362i)15-s + (−1.25 − 2.16i)16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.958 + 0.283i$
Motivic weight: \(3\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.958 + 0.283i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.595667 - 0.0862405i\)
\(L(\frac12)\) \(\approx\) \(0.595667 - 0.0862405i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.41 - 2.73i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (2.62 + 4.54i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-2.03 + 3.51i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.57 - 13.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (24.2 - 42.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 - 109.T + 6.85e3T^{2} \)
23 \( 1 + (92.3 - 159. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (47.0 + 81.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (67.6 - 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 149.T + 5.06e4T^{2} \)
41 \( 1 + (-148. + 258. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-193. - 334. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-36.7 - 63.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 633.T + 1.48e5T^{2} \)
59 \( 1 + (-162. + 281. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-34.7 - 60.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-139. + 242. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 497.T + 3.57e5T^{2} \)
73 \( 1 - 457.T + 3.89e5T^{2} \)
79 \( 1 + (-548. - 949. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (39.3 + 68.1i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 292.T + 7.04e5T^{2} \)
97 \( 1 + (82.2 + 142. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.14124237641601344146738069917, −12.51316281100083766206037504039, −11.92273335802894774819534854818, −11.08203830783778785718090569032, −9.652881853937575656186677574321, −9.446710253579614606510826025486, −7.60439851445342707334466116869, −5.22458749522162067672282152779, −3.60995909498732119252967365325, −1.40194691810868791003134941779, 0.72934243061518344713663861411, 5.18055829583864968544867478615, 6.13348045420651736013154052409, 7.33239869323674536800375526146, 8.094106472228157264954862897435, 9.826865963430766292835066998876, 10.65377223898582739887633954459, 12.34071581028708723557887157039, 13.92160965115268528224378132479, 14.62408936246082079067695531786

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