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Properties

Label 2-63-63.25-c3-0-5
Degree $2$
Conductor $63$
Sign $0.0526 - 0.998i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.30·2-s + (0.221 + 5.19i)3-s + 10.5·4-s + (7.99 − 13.8i)5-s + (−0.955 − 22.3i)6-s + (1.84 + 18.4i)7-s − 10.9·8-s + (−26.9 + 2.30i)9-s + (−34.4 + 59.6i)10-s + (6.17 + 10.6i)11-s + (2.33 + 54.7i)12-s + (35.8 + 62.0i)13-s + (−7.94 − 79.3i)14-s + (73.6 + 38.4i)15-s − 37.2·16-s + (−42.2 + 73.1i)17-s + ⋯
L(s)  = 1  − 1.52·2-s + (0.0426 + 0.999i)3-s + 1.31·4-s + (0.715 − 1.23i)5-s + (−0.0649 − 1.52i)6-s + (0.0996 + 0.995i)7-s − 0.483·8-s + (−0.996 + 0.0853i)9-s + (−1.08 + 1.88i)10-s + (0.169 + 0.293i)11-s + (0.0562 + 1.31i)12-s + (0.764 + 1.32i)13-s + (−0.151 − 1.51i)14-s + (1.26 + 0.661i)15-s − 0.581·16-s + (−0.602 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0526 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.0526 - 0.998i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.0526 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.519289 + 0.492639i\)
\(L(\frac12)\) \(\approx\) \(0.519289 + 0.492639i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.221 - 5.19i)T \)
7 \( 1 + (-1.84 - 18.4i)T \)
good2 \( 1 + 4.30T + 8T^{2} \)
5 \( 1 + (-7.99 + 13.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-6.17 - 10.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-35.8 - 62.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (42.2 - 73.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-13.6 - 23.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-20.2 + 35.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (99.2 - 171. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 292.T + 2.97e4T^{2} \)
37 \( 1 + (-58.7 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (18.6 + 32.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-122. + 212. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 91.6T + 1.03e5T^{2} \)
53 \( 1 + (-85.8 + 148. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 102.T + 2.05e5T^{2} \)
61 \( 1 + 581.T + 2.26e5T^{2} \)
67 \( 1 + 103.T + 3.00e5T^{2} \)
71 \( 1 + 204.T + 3.57e5T^{2} \)
73 \( 1 + (-580. + 1.00e3i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 621.T + 4.93e5T^{2} \)
83 \( 1 + (-67.9 + 117. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (710. + 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (559. - 968. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.22030263720670349967735419010, −13.70283206327272448404983108912, −12.13627298667830207086766801049, −10.92745511175811650799612918595, −9.699244655404077523839116866162, −8.918465614567416046520137742419, −8.508597685547675949192956631292, −6.17660880891916486962862371340, −4.62615485442701013277328233666, −1.77743823197123407183385581255, 0.849516194654313501410334470190, 2.70659843906011554366665370171, 6.23579513426844945508526342795, 7.21065230458724688149438316950, 8.094048698342304044882626064689, 9.581449584420043491622761734073, 10.69354313961552173422988472849, 11.34033965912845050210180546212, 13.40605622889485772209669887196, 13.91706801005059499633099032884

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