Properties

Label 2-63-7.2-c5-0-3
Degree $2$
Conductor $63$
Sign $-0.431 - 0.902i$
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  + (−5.31 + 9.19i)2-s + (−40.4 − 70.0i)4-s + (33.7 − 58.4i)5-s + (−120. + 48.3i)7-s + 518.·8-s + (358. + 620. i)10-s + (261. + 452. i)11-s + 76.6·13-s + (193. − 1.36e3i)14-s + (−1.46e3 + 2.53e3i)16-s + (634. + 1.09e3i)17-s + (946. − 1.63e3i)19-s − 5.45e3·20-s − 5.55e3·22-s + (−575. + 995. i)23-s + ⋯
L(s)  = 1  + (−0.938 + 1.62i)2-s + (−1.26 − 2.18i)4-s + (0.603 − 1.04i)5-s + (−0.927 + 0.373i)7-s + 2.86·8-s + (1.13 + 1.96i)10-s + (0.651 + 1.12i)11-s + 0.125·13-s + (0.264 − 1.85i)14-s + (−1.42 + 2.47i)16-s + (0.532 + 0.922i)17-s + (0.601 − 1.04i)19-s − 3.04·20-s − 2.44·22-s + (−0.226 + 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.431 - 0.902i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.511977 + 0.812021i\)
\(L(\frac12)\) \(\approx\) \(0.511977 + 0.812021i\)
\(L(\frac{7}{2})\) 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 \( 1 + (120. - 48.3i)T \)
good2 \( 1 + (5.31 - 9.19i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (-33.7 + 58.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-261. - 452. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 76.6T + 3.71e5T^{2} \)
17 \( 1 + (-634. - 1.09e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-946. + 1.63e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (575. - 995. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 3.85e3T + 2.05e7T^{2} \)
31 \( 1 + (-5.20e3 - 9.01e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-2.80e3 + 4.85e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.42e4T + 1.15e8T^{2} \)
43 \( 1 + 1.48e4T + 1.47e8T^{2} \)
47 \( 1 + (5.77e3 - 1.00e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (2.33e3 + 4.05e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.45e4 - 2.52e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (5.92e3 - 1.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.81e4 - 3.13e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.34e4T + 1.80e9T^{2} \)
73 \( 1 + (1.30e3 + 2.25e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.91e4 + 6.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.67e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.83e4 + 3.18e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.61e4T + 8.58e9T^{2} \)
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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

−14.69224272701816078196723551041, −13.50728749473909027221021674900, −12.39980819437300246411606326622, −10.07286699652073792609285388436, −9.368453490530290319742816124794, −8.545271457428020632198469267393, −7.05379802971351711302089765125, −6.00796439260602000888974011442, −4.82456766904081632903964068600, −1.16674782002206560927525559365, 0.77828172390399817671959658371, 2.68990965412730396431462304738, 3.63249163008738492953872051425, 6.40625296188008433111614921824, 8.043613984230481070182008150408, 9.554696161023303297789423349659, 10.08274458155990182045848125990, 11.15605416354272914481892787890, 12.08423417538091673258830864180, 13.45569712901278076395655333175

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