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Properties

Label 2-63-63.58-c3-0-6
Degree $2$
Conductor $63$
Sign $0.993 - 0.112i$
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  − 1.80·2-s + (−5.15 − 0.671i)3-s − 4.72·4-s + (1.04 + 1.81i)5-s + (9.32 + 1.21i)6-s + (18.3 − 2.47i)7-s + 23.0·8-s + (26.0 + 6.91i)9-s + (−1.89 − 3.28i)10-s + (−11.7 + 20.3i)11-s + (24.3 + 3.17i)12-s + (27.8 − 48.2i)13-s + (−33.2 + 4.47i)14-s + (−4.18 − 10.0i)15-s − 3.83·16-s + (55.3 + 95.9i)17-s + ⋯
L(s)  = 1  − 0.639·2-s + (−0.991 − 0.129i)3-s − 0.590·4-s + (0.0938 + 0.162i)5-s + (0.634 + 0.0826i)6-s + (0.991 − 0.133i)7-s + 1.01·8-s + (0.966 + 0.256i)9-s + (−0.0600 − 0.103i)10-s + (−0.321 + 0.557i)11-s + (0.585 + 0.0763i)12-s + (0.594 − 1.02i)13-s + (−0.633 + 0.0854i)14-s + (−0.0720 − 0.173i)15-s − 0.0599·16-s + (0.790 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.993 - 0.112i$
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: \((2,\ 63,\ (\ :3/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.742886 + 0.0419680i\)
\(L(\frac12)\) \(\approx\) \(0.742886 + 0.0419680i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.15 + 0.671i)T \)
7 \( 1 + (-18.3 + 2.47i)T \)
good2 \( 1 + 1.80T + 8T^{2} \)
5 \( 1 + (-1.04 - 1.81i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (11.7 - 20.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-27.8 + 48.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-55.3 - 95.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-9.75 + 16.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.87 - 11.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-59.9 - 103. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 158.T + 2.97e4T^{2} \)
37 \( 1 + (79.4 - 137. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-208. + 361. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-131. - 227. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 190.T + 1.03e5T^{2} \)
53 \( 1 + (-175. - 304. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 127.T + 2.05e5T^{2} \)
61 \( 1 - 724.T + 2.26e5T^{2} \)
67 \( 1 + 998.T + 3.00e5T^{2} \)
71 \( 1 + 404.T + 3.57e5T^{2} \)
73 \( 1 + (120. + 208. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 921.T + 4.93e5T^{2} \)
83 \( 1 + (502. + 869. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (239. - 414. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-431. - 747. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.50050853179063095052026452571, −13.21375923280241111822502633804, −12.21177910180369515647918697211, −10.65178900040153196978496550877, −10.28764668335758007401081411596, −8.482248176824669912464336814090, −7.49124283274104907283702254257, −5.69485985614514897478597690141, −4.45049611783759802610937557743, −1.15202886087397254050796774718, 1.03961112999360261983138910160, 4.44626309316455525209077286331, 5.54562652822311118621620033693, 7.36719120335381502643752565585, 8.693939259089492761483321205106, 9.827730953297592919706154449771, 11.03610103469832368641209570961, 11.87656880237088350197185416169, 13.37678257432583287755324251536, 14.32309617584305595871539333063

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