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Properties

Label 2-63-63.58-c3-0-15
Degree $2$
Conductor $63$
Sign $0.835 + 0.549i$
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  + 5.03·2-s + (−3.52 − 3.82i)3-s + 17.3·4-s + (−0.0751 − 0.130i)5-s + (−17.7 − 19.2i)6-s + (12.4 − 13.7i)7-s + 46.8·8-s + (−2.20 + 26.9i)9-s + (−0.378 − 0.654i)10-s + (−23.4 + 40.6i)11-s + (−60.9 − 66.1i)12-s + (−8.75 + 15.1i)13-s + (62.6 − 68.9i)14-s + (−0.232 + 0.745i)15-s + 97.4·16-s + (−35.7 − 61.9i)17-s + ⋯
L(s)  = 1  + 1.77·2-s + (−0.677 − 0.735i)3-s + 2.16·4-s + (−0.00672 − 0.0116i)5-s + (−1.20 − 1.30i)6-s + (0.672 − 0.740i)7-s + 2.07·8-s + (−0.0816 + 0.996i)9-s + (−0.0119 − 0.0207i)10-s + (−0.643 + 1.11i)11-s + (−1.46 − 1.59i)12-s + (−0.186 + 0.323i)13-s + (1.19 − 1.31i)14-s + (−0.00400 + 0.0128i)15-s + 1.52·16-s + (−0.509 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.835 + 0.549i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.91182 - 0.871149i\)
\(L(\frac12)\) \(\approx\) \(2.91182 - 0.871149i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.52 + 3.82i)T \)
7 \( 1 + (-12.4 + 13.7i)T \)
good2 \( 1 - 5.03T + 8T^{2} \)
5 \( 1 + (0.0751 + 0.130i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (23.4 - 40.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (8.75 - 15.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (35.7 + 61.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (57.4 - 99.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-67.6 - 117. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (30.1 + 52.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 10.2T + 2.97e4T^{2} \)
37 \( 1 + (-152. + 263. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (142. - 247. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (234. + 406. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 337.T + 1.03e5T^{2} \)
53 \( 1 + (-13.1 - 22.7i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 579.T + 2.05e5T^{2} \)
61 \( 1 - 807.T + 2.26e5T^{2} \)
67 \( 1 + 121.T + 3.00e5T^{2} \)
71 \( 1 + 26.5T + 3.57e5T^{2} \)
73 \( 1 + (60.9 + 105. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 58.6T + 4.93e5T^{2} \)
83 \( 1 + (-298. - 517. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-286. + 495. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (335. + 581. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.08009068481884465583976280146, −13.24955680834023562323525484947, −12.35270041265192917716666806784, −11.47566794448332711379204910656, −10.44126447074562256331130353216, −7.64119675120668240157477014405, −6.79026510256423410300875784663, −5.33345575362914670751584935767, −4.36140342262414739459850512934, −2.09327730470527907730867765456, 2.93127463033099680006752108083, 4.58292944098047254772081253773, 5.45506343647842953051745908973, 6.53319114267198669700908731769, 8.648787128758295418148657278606, 10.80288554917389289687423533722, 11.25158032495011507045466644293, 12.47673187793674159691985611158, 13.35969643270486143114007155301, 14.87459532324674932546327410179

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