Properties

Degree $2$
Conductor $63$
Sign $-0.994 - 0.102i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 2.26i)2-s + (−3.13 − 4.14i)3-s + (0.578 − 1.00i)4-s + (6.77 − 11.7i)5-s + (−5.29 + 12.5i)6-s + (3.5 + 6.06i)7-s − 23.9·8-s + (−7.38 + 25.9i)9-s − 35.4·10-s + (−12.0 − 20.8i)11-s + (−5.97 + 0.740i)12-s + (−11.9 + 20.7i)13-s + (9.15 − 15.8i)14-s + (−69.8 + 8.65i)15-s + (26.6 + 46.2i)16-s + 79.6·17-s + ⋯
L(s)  = 1  + (−0.462 − 0.800i)2-s + (−0.602 − 0.797i)3-s + (0.0723 − 0.125i)4-s + (0.605 − 1.04i)5-s + (−0.360 + 0.851i)6-s + (0.188 + 0.327i)7-s − 1.05·8-s + (−0.273 + 0.961i)9-s − 1.12·10-s + (−0.329 − 0.571i)11-s + (−0.143 + 0.0178i)12-s + (−0.255 + 0.443i)13-s + (0.174 − 0.302i)14-s + (−1.20 + 0.149i)15-s + (0.417 + 0.722i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.994 - 0.102i$
Motivic weight: \(3\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.994 - 0.102i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0459966 + 0.897587i\)
\(L(\frac12)\) \(\approx\) \(0.0459966 + 0.897587i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.13 + 4.14i)T \)
7 \( 1 + (-3.5 - 6.06i)T \)
good2 \( 1 + (1.30 + 2.26i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-6.77 + 11.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (12.0 + 20.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (11.9 - 20.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 79.6T + 4.91e3T^{2} \)
19 \( 1 + 50.0T + 6.85e3T^{2} \)
23 \( 1 + (-75.8 + 131. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (128. + 221. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-1.36 + 2.35i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 319.T + 5.06e4T^{2} \)
41 \( 1 + (82.3 - 142. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (211. + 366. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (50.0 + 86.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 194.T + 1.48e5T^{2} \)
59 \( 1 + (288. - 499. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-21.5 - 37.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-508. + 880. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 509.T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 + (-447. - 774. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (7.23 + 12.5i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.53e3T + 7.04e5T^{2} \)
97 \( 1 + (-887. - 1.53e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.47472419728011658154931505985, −12.51525663910791696674221109813, −11.69951354727673157335042313656, −10.58603128531529212843068056236, −9.341458390105325038185056945038, −8.179260834468491612434185389933, −6.25350300107881933562215489311, −5.22779504808598686552317728610, −2.20914620181221725350415928445, −0.77697921606178826144602194153, 3.23122089043815030542848849651, 5.38646412467684909060179676768, 6.60195236572471443444948687745, 7.68558168809486539726851932402, 9.399720314089721424201044767764, 10.32734117487245413618678251546, 11.38178570279641547097385629405, 12.75368197130375891601843964708, 14.60075483726366558403420960039, 15.02077402989039640099001698524

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