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Properties

Label 2-63-63.38-c3-0-21
Degree $2$
Conductor $63$
Sign $-0.152 - 0.988i$
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  − 3.64i·2-s + (−4.97 − 1.49i)3-s − 5.32·4-s + (−6.11 + 10.5i)5-s + (−5.46 + 18.1i)6-s + (−17.8 + 4.79i)7-s − 9.77i·8-s + (22.5 + 14.9i)9-s + (38.6 + 22.3i)10-s + (8.67 − 5.00i)11-s + (26.4 + 7.97i)12-s + (−57.0 + 32.9i)13-s + (17.5 + 65.2i)14-s + (46.3 − 43.5i)15-s − 78.2·16-s + (38.6 − 66.9i)17-s + ⋯
L(s)  = 1  − 1.29i·2-s + (−0.957 − 0.288i)3-s − 0.665·4-s + (−0.547 + 0.947i)5-s + (−0.372 + 1.23i)6-s + (−0.965 + 0.259i)7-s − 0.432i·8-s + (0.833 + 0.552i)9-s + (1.22 + 0.705i)10-s + (0.237 − 0.137i)11-s + (0.636 + 0.191i)12-s + (−1.21 + 0.702i)13-s + (0.334 + 1.24i)14-s + (0.796 − 0.749i)15-s − 1.22·16-s + (0.551 − 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.152 - 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00266942 + 0.00311201i\)
\(L(\frac12)\) \(\approx\) \(0.00266942 + 0.00311201i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.97 + 1.49i)T \)
7 \( 1 + (17.8 - 4.79i)T \)
good2 \( 1 + 3.64iT - 8T^{2} \)
5 \( 1 + (6.11 - 10.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-8.67 + 5.00i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (57.0 - 32.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-38.6 + 66.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (104. - 60.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (106. + 61.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (50.3 + 29.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 201. iT - 2.97e4T^{2} \)
37 \( 1 + (-82.9 - 143. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-112. - 194. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 22.5T + 1.03e5T^{2} \)
53 \( 1 + (360. + 207. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 360.T + 2.05e5T^{2} \)
61 \( 1 - 734. iT - 2.26e5T^{2} \)
67 \( 1 + 111.T + 3.00e5T^{2} \)
71 \( 1 - 455. iT - 3.57e5T^{2} \)
73 \( 1 + (662. + 382. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 43.1T + 4.93e5T^{2} \)
83 \( 1 + (-200. + 346. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (347. + 602. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-955. - 551. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.11879196659716348821864683457, −12.09939314586188126938386205753, −11.60447612304458684286914060032, −10.42603767929567387948678673257, −9.663759674867942162079202254190, −7.30503959681295571036867458804, −6.28738167405950947981880739887, −4.11032710863425628600978412632, −2.46645073191298554755740563286, −0.00297362082178684213804792000, 4.32547361450236736465696941068, 5.55828614610662130967341762410, 6.69582981293339713669275396432, 7.88843867587040480912798839432, 9.325352200600843921923868142297, 10.65438226379373468091640007022, 12.23815541132972691526673847474, 12.84551920513474589349629404284, 14.63292101912803040072720459600, 15.64564222266123462098134397280

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