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Properties

Label 1-463-463.49-r0-0-0
Degree $1$
Conductor $463$
Sign $0.799 - 0.600i$
Analytic cond. $2.15016$
Root an. cond. $2.15016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.992 − 0.122i)2-s + (−0.523 − 0.852i)3-s + (0.970 + 0.242i)4-s + (0.262 − 0.965i)5-s + (0.415 + 0.909i)6-s + (−0.523 + 0.852i)7-s + (−0.933 − 0.359i)8-s + (−0.452 + 0.891i)9-s + (−0.377 + 0.925i)10-s + (−0.992 + 0.122i)11-s + (−0.301 − 0.953i)12-s + (−0.900 + 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.959 + 0.281i)15-s + (0.882 + 0.470i)16-s + (0.947 + 0.320i)17-s + ⋯
L(s)  = 1  + (−0.992 − 0.122i)2-s + (−0.523 − 0.852i)3-s + (0.970 + 0.242i)4-s + (0.262 − 0.965i)5-s + (0.415 + 0.909i)6-s + (−0.523 + 0.852i)7-s + (−0.933 − 0.359i)8-s + (−0.452 + 0.891i)9-s + (−0.377 + 0.925i)10-s + (−0.992 + 0.122i)11-s + (−0.301 − 0.953i)12-s + (−0.900 + 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.959 + 0.281i)15-s + (0.882 + 0.470i)16-s + (0.947 + 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 463 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(463\)
Sign: $0.799 - 0.600i$
Analytic conductor: \(2.15016\)
Root analytic conductor: \(2.15016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{463} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 463,\ (0:\ ),\ 0.799 - 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5564680866 - 0.1856209000i\)
\(L(\frac12)\) \(\approx\) \(0.5564680866 - 0.1856209000i\)
\(L(1)\) \(\approx\) \(0.5422349897 - 0.1596642194i\)
\(L(1)\) \(\approx\) \(0.5422349897 - 0.1596642194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad463 \( 1 \)
good2 \( 1 + (-0.992 - 0.122i)T \)
3 \( 1 + (-0.523 - 0.852i)T \)
5 \( 1 + (0.262 - 0.965i)T \)
7 \( 1 + (-0.523 + 0.852i)T \)
11 \( 1 + (-0.992 + 0.122i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (0.947 + 0.320i)T \)
19 \( 1 + (0.742 + 0.670i)T \)
23 \( 1 + (0.488 - 0.872i)T \)
29 \( 1 + (0.882 - 0.470i)T \)
31 \( 1 + (0.917 + 0.396i)T \)
37 \( 1 + (-0.862 + 0.505i)T \)
41 \( 1 + (0.182 + 0.983i)T \)
43 \( 1 + (0.488 - 0.872i)T \)
47 \( 1 + (0.986 + 0.162i)T \)
53 \( 1 + (-0.979 - 0.202i)T \)
59 \( 1 + (0.101 - 0.994i)T \)
61 \( 1 + (0.986 + 0.162i)T \)
67 \( 1 + (0.557 - 0.830i)T \)
71 \( 1 + (0.101 + 0.994i)T \)
73 \( 1 + (-0.818 + 0.574i)T \)
79 \( 1 + (-0.301 + 0.953i)T \)
83 \( 1 + (0.947 - 0.320i)T \)
89 \( 1 + (0.947 + 0.320i)T \)
97 \( 1 + (0.262 + 0.965i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.855347876004577880037845171434, −23.13045729863836585652214156697, −22.29925852714163270282968600429, −21.30605485867322506317225074479, −20.59976234360491403446279243253, −19.58562178658808943005757917396, −18.81528615780569360428021693508, −17.55571895158467974465915744269, −17.46945898643632207458540100831, −16.17576623043668794337041216778, −15.6788434691083116273085263728, −14.74232906278741776207239525377, −13.76441949963697569287371806388, −12.270555582465454576887416027476, −11.25131300152172343531249927854, −10.4041495122476183705418976962, −10.06508836858449834798238640950, −9.22740247570019617254167328491, −7.64504181760322881162505938047, −7.10114734033138555555246201319, −5.95971349602061253323152756362, −5.07293146704851157595803428935, −3.33883791122445051159418743408, −2.72914261719423232925840222991, −0.70573964397583178321434896014, 0.820506844708984709143062349861, 2.01895978909166481043757225231, 2.87660234218842778529609076276, 5.00752319042842587102625537838, 5.80311953830831634531970673881, 6.7741174878189391081650598891, 7.90828408504670224677736158665, 8.48433446157300749191346658522, 9.66465409250126937022200090240, 10.334493671695950030198153844314, 11.77224736170764640070504093278, 12.319456386705035230843271084824, 12.83390697746140373515700166556, 14.17099637756538301547498682977, 15.6285620861293007385842998032, 16.30938944251541359523494685027, 17.08739674001029845356612085105, 17.78452725524224528851925228937, 18.88161364474890366977265780720, 19.06717493927223587376143970757, 20.27702457560195483391704571322, 21.09507735879759167337041404780, 21.96338128704046994628175181961, 23.181263092986574007519291990736, 24.09707557700181552539512773531

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