Properties

Label 2-63-21.17-c5-0-5
Degree $2$
Conductor $63$
Sign $0.911 + 0.410i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.14 + 4.70i)2-s + (28.2 − 48.8i)4-s + (33.1 + 57.4i)5-s + (−116. − 57.3i)7-s + 229. i·8-s + (−540. − 312. i)10-s + (−383. − 221. i)11-s − 437. i·13-s + (1.21e3 − 79.6i)14-s + (−176. − 306. i)16-s + (−476. + 825. i)17-s + (1.93e3 − 1.11e3i)19-s + 3.74e3·20-s + 4.16e3·22-s + (2.09e3 − 1.21e3i)23-s + ⋯
L(s)  = 1  + (−1.43 + 0.831i)2-s + (0.881 − 1.52i)4-s + (0.593 + 1.02i)5-s + (−0.896 − 0.442i)7-s + 1.26i·8-s + (−1.70 − 0.986i)10-s + (−0.955 − 0.551i)11-s − 0.717i·13-s + (1.65 − 0.108i)14-s + (−0.172 − 0.299i)16-s + (−0.399 + 0.692i)17-s + (1.22 − 0.709i)19-s + 2.09·20-s + 1.83·22-s + (0.826 − 0.477i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.911 + 0.410i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.550995 - 0.118392i\)
\(L(\frac12)\) \(\approx\) \(0.550995 - 0.118392i\)
\(L(\frac{7}{2})\) 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 \( 1 + (116. + 57.3i)T \)
good2 \( 1 + (8.14 - 4.70i)T + (16 - 27.7i)T^{2} \)
5 \( 1 + (-33.1 - 57.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (383. + 221. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 437. iT - 3.71e5T^{2} \)
17 \( 1 + (476. - 825. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.93e3 + 1.11e3i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.09e3 + 1.21e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 6.38e3iT - 2.05e7T^{2} \)
31 \( 1 + (-5.91e3 - 3.41e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (7.60e3 + 1.31e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.72e3T + 1.15e8T^{2} \)
43 \( 1 - 1.37e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.07e3 - 1.57e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (2.07e4 + 1.20e4i)T + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.12e4 + 3.67e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.77e4 - 1.02e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (290. - 503. i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.26e4iT - 1.80e9T^{2} \)
73 \( 1 + (4.74e4 + 2.73e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.85e4 + 3.22e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 4.46e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.20e4 - 2.09e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.89e4iT - 8.58e9T^{2} \)
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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

−14.12934336169496032236750637397, −13.01776400760230001613375433686, −10.81685523604123532839789644432, −10.30593000441801220258838009876, −9.246399694137609014856863158908, −7.84532410428670353562415926658, −6.81721013977859308346619370925, −5.85150380611362600938401588958, −2.86236506612957415798512362495, −0.46143508320704541986290625631, 1.26915683261486615390277835769, 2.79522705724784153549084617338, 5.25225157442924303283487723560, 7.21548210239666547250581042161, 8.697210701235581703565048549929, 9.447621938330221371235494982588, 10.19263433675177282728271865505, 11.70811502509274091719919149964, 12.58271152812529163764856785804, 13.62588915613647579896248671258

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