Properties

Degree $2$
Conductor $63$
Sign $-0.986 + 0.166i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.21 − 2.43i)2-s + (7.83 + 13.5i)4-s + (6.38 − 11.0i)5-s + (2.53 − 18.3i)7-s − 37.3i·8-s + (−53.7 + 31.0i)10-s + (−46.8 + 27.0i)11-s − 8.85i·13-s + (−55.3 + 71.1i)14-s + (−28.1 + 48.7i)16-s + (−34.4 − 59.6i)17-s + (−141. − 81.9i)19-s + 200.·20-s + 263.·22-s + (81.3 + 46.9i)23-s + ⋯
L(s)  = 1  + (−1.48 − 0.860i)2-s + (0.979 + 1.69i)4-s + (0.570 − 0.988i)5-s + (0.137 − 0.990i)7-s − 1.64i·8-s + (−1.70 + 0.981i)10-s + (−1.28 + 0.741i)11-s − 0.188i·13-s + (−1.05 + 1.35i)14-s + (−0.439 + 0.760i)16-s + (−0.491 − 0.851i)17-s + (−1.71 − 0.989i)19-s + 2.23·20-s + 2.55·22-s + (0.737 + 0.425i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.986 + 0.166i$
Motivic weight: \(3\)
Character: $\chi_{63} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.986 + 0.166i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0436034 - 0.519598i\)
\(L(\frac12)\) \(\approx\) \(0.0436034 - 0.519598i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.53 + 18.3i)T \)
good2 \( 1 + (4.21 + 2.43i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-6.38 + 11.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (46.8 - 27.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 8.85iT - 2.19e3T^{2} \)
17 \( 1 + (34.4 + 59.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (141. + 81.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-81.3 - 46.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 119. iT - 2.43e4T^{2} \)
31 \( 1 + (-85.6 + 49.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (47.0 - 81.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 259.T + 6.89e4T^{2} \)
43 \( 1 - 5.01T + 7.95e4T^{2} \)
47 \( 1 + (-28.6 + 49.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-407. + 235. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (112. + 195. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-370. - 213. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (81.9 + 141. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 79.8iT - 3.57e5T^{2} \)
73 \( 1 + (-666. + 384. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (267. - 463. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 438.T + 5.71e5T^{2} \)
89 \( 1 + (-12.8 + 22.2i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.38e3iT - 9.12e5T^{2} \)
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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

−13.39396503301620769708473180575, −12.75726743123957140173543626361, −11.24257613284277506417899994034, −10.33899323602411772075569016999, −9.411938691065426556031642710385, −8.339159010389600737074253395755, −7.17913494228172865148648129402, −4.76965275029684029995942088502, −2.30128925902671239552605327053, −0.55163673812741434068905960815, 2.32335623688397647169209057820, 5.76901447786719381863348568346, 6.60542105266445108436850495526, 8.121491496787669222158176474888, 8.902204183870537655186271756957, 10.40908539816289060554350280605, 10.82289276232096370019334097340, 12.80794758880535556082445851407, 14.47939182931000055967575090263, 15.17935013043170345317982731104

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