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Properties

Label 2-63-63.5-c3-0-12
Degree $2$
Conductor $63$
Sign $0.857 + 0.514i$
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  − 0.837i·2-s + (4.56 + 2.47i)3-s + 7.29·4-s + (−10.9 − 18.9i)5-s + (2.07 − 3.82i)6-s + (14.8 + 11.0i)7-s − 12.8i·8-s + (14.7 + 22.6i)9-s + (−15.8 + 9.17i)10-s + (13.3 + 7.73i)11-s + (33.3 + 18.0i)12-s + (−33.3 − 19.2i)13-s + (9.29 − 12.4i)14-s + (−3.11 − 113. i)15-s + 47.6·16-s + (34.8 + 60.3i)17-s + ⋯
L(s)  = 1  − 0.296i·2-s + (0.879 + 0.476i)3-s + 0.912·4-s + (−0.980 − 1.69i)5-s + (0.140 − 0.260i)6-s + (0.800 + 0.599i)7-s − 0.566i·8-s + (0.546 + 0.837i)9-s + (−0.502 + 0.290i)10-s + (0.367 + 0.211i)11-s + (0.802 + 0.434i)12-s + (−0.711 − 0.410i)13-s + (0.177 − 0.237i)14-s + (−0.0535 − 1.95i)15-s + 0.744·16-s + (0.497 + 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.94708 - 0.538907i\)
\(L(\frac12)\) \(\approx\) \(1.94708 - 0.538907i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.56 - 2.47i)T \)
7 \( 1 + (-14.8 - 11.0i)T \)
good2 \( 1 + 0.837iT - 8T^{2} \)
5 \( 1 + (10.9 + 18.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-13.3 - 7.73i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (33.3 + 19.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-34.8 - 60.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (55.4 + 32.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (60.4 - 34.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (168. - 97.3i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 78.6iT - 2.97e4T^{2} \)
37 \( 1 + (-3.34 + 5.79i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (9.21 - 15.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (12.2 + 21.1i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 276.T + 1.03e5T^{2} \)
53 \( 1 + (-95.3 + 55.0i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 353.T + 2.05e5T^{2} \)
61 \( 1 + 531. iT - 2.26e5T^{2} \)
67 \( 1 - 524.T + 3.00e5T^{2} \)
71 \( 1 + 43.3iT - 3.57e5T^{2} \)
73 \( 1 + (-54.9 + 31.7i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.21e3T + 4.93e5T^{2} \)
83 \( 1 + (111. + 192. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-35.2 + 61.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (483. - 279. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.81634451657477976938693938331, −12.90473584754240181986584606688, −12.20325575952183315746284479996, −11.13573951712602129974077985982, −9.575872451621033220730522103909, −8.398866243496152828830062679945, −7.67374490014079282676918894517, −5.19244800946295656969958705301, −3.86441187326825634666849522477, −1.79011687185698211324932681827, 2.35712774676350650156067847673, 3.76262922327367067546744189931, 6.59746808746498981703841763953, 7.37161972648819611360307428035, 8.049817798929453365948118447355, 10.16168747613688652099121866589, 11.30304097095731300194306887951, 12.02393681646744224014815663389, 14.05651639103700803048390104972, 14.61192003139352490369500233834

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