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Properties

Label 2-63-63.58-c3-0-7
Degree $2$
Conductor $63$
Sign $0.312 + 0.949i$
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  − 2.93·2-s + (−3.89 + 3.43i)3-s + 0.611·4-s + (1.84 + 3.20i)5-s + (11.4 − 10.0i)6-s + (−18.1 + 3.79i)7-s + 21.6·8-s + (3.41 − 26.7i)9-s + (−5.42 − 9.39i)10-s + (32.8 − 56.9i)11-s + (−2.38 + 2.10i)12-s + (2.73 − 4.74i)13-s + (53.1 − 11.1i)14-s + (−18.2 − 6.13i)15-s − 68.5·16-s + (−25.1 − 43.5i)17-s + ⋯
L(s)  = 1  − 1.03·2-s + (−0.750 + 0.660i)3-s + 0.0764·4-s + (0.165 + 0.286i)5-s + (0.778 − 0.685i)6-s + (−0.978 + 0.204i)7-s + 0.958·8-s + (0.126 − 0.991i)9-s + (−0.171 − 0.297i)10-s + (0.900 − 1.55i)11-s + (−0.0573 + 0.0505i)12-s + (0.0584 − 0.101i)13-s + (1.01 − 0.212i)14-s + (−0.313 − 0.105i)15-s − 1.07·16-s + (−0.358 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.312 + 0.949i$
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: \((2,\ 63,\ (\ :3/2),\ 0.312 + 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.298103 - 0.215807i\)
\(L(\frac12)\) \(\approx\) \(0.298103 - 0.215807i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.89 - 3.43i)T \)
7 \( 1 + (18.1 - 3.79i)T \)
good2 \( 1 + 2.93T + 8T^{2} \)
5 \( 1 + (-1.84 - 3.20i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-32.8 + 56.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-2.73 + 4.74i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (25.1 + 43.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-0.769 + 1.33i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-60.0 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (39.2 + 68.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 303.T + 2.97e4T^{2} \)
37 \( 1 + (-96.6 + 167. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-196. + 340. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (138. + 239. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 252.T + 1.03e5T^{2} \)
53 \( 1 + (204. + 353. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 262.T + 2.05e5T^{2} \)
61 \( 1 + 112.T + 2.26e5T^{2} \)
67 \( 1 + 98.2T + 3.00e5T^{2} \)
71 \( 1 + 255.T + 3.57e5T^{2} \)
73 \( 1 + (-344. - 596. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.08e3T + 4.93e5T^{2} \)
83 \( 1 + (-152. - 263. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-550. + 953. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-493. - 855. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.32209687600434644363695594566, −13.09366555219952514129459119931, −11.50957826348356499530760334654, −10.66524283873360916599599476925, −9.482130117279359145662819823962, −8.871172880775739923071451892469, −6.92601378771083517265341992411, −5.65432215783656694242194665426, −3.67767638773258152662984851699, −0.42383459631637017050584400863, 1.47885114761465273875488381698, 4.60686671160693347115939154098, 6.51647590636399518424799552649, 7.43540480281205316763240482683, 9.023962823011078900648049610566, 9.935246280835458254798123303993, 11.08763329117623399823928631542, 12.63038457490846215728994724212, 13.11202716802771179071611907442, 14.73458165473509656091133699508

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