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Properties

Label 2-63-7.4-c5-0-10
Degree $2$
Conductor $63$
Sign $0.994 - 0.102i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 2.37i)2-s + (12.2 − 21.1i)4-s + (29.1 + 50.5i)5-s + (−21.4 − 127. i)7-s + 155.·8-s + (−80.2 + 138. i)10-s + (8.71 − 15.0i)11-s + 889.·13-s + (274. − 226. i)14-s + (−178. − 308. i)16-s + (513. − 889. i)17-s + (869. + 1.50e3i)19-s + 1.42e3·20-s + 47.8·22-s + (1.96e3 + 3.40e3i)23-s + ⋯
L(s)  = 1  + (0.242 + 0.420i)2-s + (0.381 − 0.661i)4-s + (0.522 + 0.904i)5-s + (−0.165 − 0.986i)7-s + 0.856·8-s + (−0.253 + 0.439i)10-s + (0.0217 − 0.0376i)11-s + 1.46·13-s + (0.374 − 0.309i)14-s + (−0.173 − 0.301i)16-s + (0.430 − 0.746i)17-s + (0.552 + 0.957i)19-s + 0.797·20-s + 0.0210·22-s + (0.775 + 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.994 - 0.102i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ 0.994 - 0.102i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.47001 + 0.126996i\)
\(L(\frac12)\) \(\approx\) \(2.47001 + 0.126996i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (21.4 + 127. i)T \)
good2 \( 1 + (-1.37 - 2.37i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-29.1 - 50.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-8.71 + 15.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 889.T + 3.71e5T^{2} \)
17 \( 1 + (-513. + 889. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-869. - 1.50e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.96e3 - 3.40e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 5.63e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.54e3 + 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.51e3 + 4.35e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.83e4T + 1.15e8T^{2} \)
43 \( 1 + 1.63e3T + 1.47e8T^{2} \)
47 \( 1 + (4.80e3 + 8.31e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.16e4 - 2.01e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.80e3 - 3.12e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.14e4 + 1.98e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.35e4 - 4.07e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.59e3T + 1.80e9T^{2} \)
73 \( 1 + (2.96e3 - 5.13e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.42e4 - 7.66e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 9.58e4T + 3.93e9T^{2} \)
89 \( 1 + (2.32e4 + 4.02e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 7.59e4T + 8.58e9T^{2} \)
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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

−13.91921655509214965507532250650, −13.48963636943716245794381606601, −11.39999928392880380861640886849, −10.57791415580590568498951403150, −9.644740063336668464470147278693, −7.58263478147098886454878801947, −6.60440072420327010918554362523, −5.51930212590954880588010106326, −3.51283288971617070275976505547, −1.38095645184881750988500101384, 1.60100842539679883761423261546, 3.25916490878785512029035979922, 5.02018787831344439120887918505, 6.46680667916689134201252454535, 8.302501559144770194127863087550, 9.105021111815970588157067465406, 10.76557484746689654554130852349, 11.88228823810789006660822442664, 12.85690485658925676281368137808, 13.44745199180996289892886574073

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