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Properties

Label 2-63-63.25-c3-0-19
Degree $2$
Conductor $63$
Sign $0.852 + 0.522i$
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  + 3.35·2-s + (3.94 − 3.37i)3-s + 3.24·4-s + (4.35 − 7.54i)5-s + (13.2 − 11.3i)6-s + (−2.88 + 18.2i)7-s − 15.9·8-s + (4.19 − 26.6i)9-s + (14.5 − 25.2i)10-s + (5.89 + 10.2i)11-s + (12.8 − 10.9i)12-s + (26.5 + 46.0i)13-s + (−9.66 + 61.3i)14-s + (−8.27 − 44.4i)15-s − 79.4·16-s + (−22.5 + 39.0i)17-s + ⋯
L(s)  = 1  + 1.18·2-s + (0.760 − 0.649i)3-s + 0.405·4-s + (0.389 − 0.674i)5-s + (0.901 − 0.770i)6-s + (−0.155 + 0.987i)7-s − 0.704·8-s + (0.155 − 0.987i)9-s + (0.461 − 0.799i)10-s + (0.161 + 0.280i)11-s + (0.308 − 0.263i)12-s + (0.567 + 0.982i)13-s + (−0.184 + 1.17i)14-s + (−0.142 − 0.765i)15-s − 1.24·16-s + (−0.321 + 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.852 + 0.522i$
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: \((2,\ 63,\ (\ :3/2),\ 0.852 + 0.522i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.78730 - 0.786462i\)
\(L(\frac12)\) \(\approx\) \(2.78730 - 0.786462i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.94 + 3.37i)T \)
7 \( 1 + (2.88 - 18.2i)T \)
good2 \( 1 - 3.35T + 8T^{2} \)
5 \( 1 + (-4.35 + 7.54i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-5.89 - 10.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-26.5 - 46.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (22.5 - 39.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (13.0 + 22.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-29.0 + 50.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-36.7 + 63.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 314.T + 2.97e4T^{2} \)
37 \( 1 + (189. + 327. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-181. - 313. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-58.7 + 101. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 228.T + 1.03e5T^{2} \)
53 \( 1 + (-307. + 532. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 576.T + 2.05e5T^{2} \)
61 \( 1 - 446.T + 2.26e5T^{2} \)
67 \( 1 - 297.T + 3.00e5T^{2} \)
71 \( 1 + 866.T + 3.57e5T^{2} \)
73 \( 1 + (-283. + 490. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 366.T + 4.93e5T^{2} \)
83 \( 1 + (510. - 884. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (247. + 427. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (76.3 - 132. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−14.27830534363669476361131465364, −13.07107477147783033460371564585, −12.71620386205038906932225768913, −11.58565319797214719070384376338, −9.218927682181036172285169611782, −8.745451537477688976423169260855, −6.72339084366682117226737702935, −5.52948598679351051817261532715, −3.91854313727928673185059089476, −2.15766313709082433433667373206, 2.99912292933211254641712905598, 4.01531161574786158728973883651, 5.52983279402122580825349350668, 7.11256144400011469507637747588, 8.783627738943090312766715474361, 10.15573550762709797576989576351, 11.06994511990027080347522295189, 12.86543891763031318215708722075, 13.73338927623495104348608293903, 14.29993525441203722898690120201

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