Properties

Degree $2$
Conductor $63$
Sign $-0.999 - 0.0204i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 21.0·5-s + (−11 + 14.8i)7-s + 22.6i·8-s − 59.5i·10-s + 15.5i·11-s − 29.7i·13-s + (−42.1 − 31.1i)14-s − 64.0·16-s + 63.2·17-s + 89.3i·19-s − 44·22-s − 77.7i·23-s + 318.·25-s + 84.2·26-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.88·5-s + (−0.593 + 0.804i)7-s + 0.999i·8-s − 1.88i·10-s + 0.426i·11-s − 0.635i·13-s + (−0.804 − 0.593i)14-s − 1.00·16-s + 0.901·17-s + 1.07i·19-s − 0.426·22-s − 0.705i·23-s + 2.55·25-s + 0.635·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0204i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.999 - 0.0204i$
Motivic weight: \(3\)
Character: $\chi_{63} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.999 - 0.0204i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00804613 + 0.785999i\)
\(L(\frac12)\) \(\approx\) \(0.00804613 + 0.785999i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (11 - 14.8i)T \)
good2 \( 1 - 2.82iT - 8T^{2} \)
5 \( 1 + 21.0T + 125T^{2} \)
11 \( 1 - 15.5iT - 1.33e3T^{2} \)
13 \( 1 + 29.7iT - 2.19e3T^{2} \)
17 \( 1 - 63.2T + 4.91e3T^{2} \)
19 \( 1 - 89.3iT - 6.85e3T^{2} \)
23 \( 1 + 77.7iT - 1.21e4T^{2} \)
29 \( 1 - 125. iT - 2.43e4T^{2} \)
31 \( 1 - 238. iT - 2.97e4T^{2} \)
37 \( 1 + 184T + 5.06e4T^{2} \)
41 \( 1 - 105.T + 6.89e4T^{2} \)
43 \( 1 + 190T + 7.95e4T^{2} \)
47 \( 1 + 42.1T + 1.03e5T^{2} \)
53 \( 1 + 357. iT - 1.48e5T^{2} \)
59 \( 1 + 84.2T + 2.05e5T^{2} \)
61 \( 1 - 655. iT - 2.26e5T^{2} \)
67 \( 1 - 296T + 3.00e5T^{2} \)
71 \( 1 - 329. iT - 3.57e5T^{2} \)
73 \( 1 + 804. iT - 3.89e5T^{2} \)
79 \( 1 - 836T + 4.93e5T^{2} \)
83 \( 1 - 1.22e3T + 5.71e5T^{2} \)
89 \( 1 + 695.T + 7.04e5T^{2} \)
97 \( 1 - 566. iT - 9.12e5T^{2} \)
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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

−15.14484154402585056385357767080, −14.56867982112399747676689084239, −12.45640310706611736288809972565, −11.96295467126381038910488315068, −10.55939556092226575416043890539, −8.615087947156589652757809777505, −7.81414153030141916024143296395, −6.74280033067996113785743085337, −5.19318646703473447541040739388, −3.32880601330551707161350680948, 0.53856234751400809301860077153, 3.26560050153144205464053973087, 4.18225306365907447239200869850, 6.84292784910064500924589121149, 7.83296546528896013648912018195, 9.522816736914264142086769371306, 10.90343172657409847469808742847, 11.54574217363783809685834175951, 12.42856316904692678483301745266, 13.60807334770364290077829177500

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