Properties

Degree $2$
Conductor $63$
Sign $0.862 - 0.505i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 − 0.910i)2-s + (−2.34 + 4.05i)4-s + (7.54 + 13.0i)5-s + (16.2 − 8.80i)7-s + 23.0i·8-s + (23.7 + 13.7i)10-s + (8.56 + 4.94i)11-s − 67.8i·13-s + (17.6 − 28.7i)14-s + (2.27 + 3.93i)16-s + (−35.0 + 60.7i)17-s + (−53.2 + 30.7i)19-s − 70.7·20-s + 18.0·22-s + (113. − 65.7i)23-s + ⋯
L(s)  = 1  + (0.557 − 0.321i)2-s + (−0.292 + 0.507i)4-s + (0.674 + 1.16i)5-s + (0.879 − 0.475i)7-s + 1.02i·8-s + (0.752 + 0.434i)10-s + (0.234 + 0.135i)11-s − 1.44i·13-s + (0.337 − 0.548i)14-s + (0.0355 + 0.0615i)16-s + (−0.500 + 0.866i)17-s + (−0.642 + 0.371i)19-s − 0.790·20-s + 0.174·22-s + (1.03 − 0.596i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.862 - 0.505i$
Motivic weight: \(3\)
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.862 - 0.505i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.91099 + 0.519098i\)
\(L(\frac12)\) \(\approx\) \(1.91099 + 0.519098i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.2 + 8.80i)T \)
good2 \( 1 + (-1.57 + 0.910i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-7.54 - 13.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-8.56 - 4.94i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 67.8iT - 2.19e3T^{2} \)
17 \( 1 + (35.0 - 60.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (53.2 - 30.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-113. + 65.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 158. iT - 2.43e4T^{2} \)
31 \( 1 + (66.2 + 38.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (174. + 301. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 138.T + 6.89e4T^{2} \)
43 \( 1 - 539.T + 7.95e4T^{2} \)
47 \( 1 + (111. + 193. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-459. - 265. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (271. - 470. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (116. - 67.0i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (160. - 277. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 416. iT - 3.57e5T^{2} \)
73 \( 1 + (-472. - 272. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-161. - 279. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 885.T + 5.71e5T^{2} \)
89 \( 1 + (812. + 1.40e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 739. iT - 9.12e5T^{2} \)
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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

−14.45754611042020139400721981353, −13.47647031936171101833299379999, −12.51871774896178816480619674564, −11.02866613049640481801075135397, −10.43052920511272424286099563125, −8.571739661729662157842767801327, −7.34244631435980309111218919017, −5.70081650457084560363137594285, −4.05034657919307656217657668826, −2.47647433328230877142271468005, 1.51051420637737272417373234747, 4.58553394522548473168608315268, 5.29086332396067042120374576269, 6.75449847002380620229600685614, 8.864976028133473370378687533655, 9.331737968934660585981334036047, 11.10703660485721962028160205982, 12.39310968018578765366886394918, 13.51301170157823363920586685985, 14.20038889526236766722753064887

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