Properties

Label 2-50-25.4-c5-0-5
Degree $2$
Conductor $50$
Sign $0.852 - 0.523i$
Analytic cond. $8.01919$
Root an. cond. $2.83181$
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  + (−3.80 − 1.23i)2-s + (9.57 − 13.1i)3-s + (12.9 + 9.40i)4-s + (−0.421 + 55.9i)5-s + (−52.7 + 38.2i)6-s + 120. i·7-s + (−37.6 − 51.7i)8-s + (−6.90 − 21.2i)9-s + (70.7 − 212. i)10-s + (−31.8 + 97.9i)11-s + (247. − 80.5i)12-s + (255. − 83.1i)13-s + (148. − 458. i)14-s + (732. + 540. i)15-s + (79.1 + 243. i)16-s + (1.17e3 + 1.61e3i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.614 − 0.845i)3-s + (0.404 + 0.293i)4-s + (−0.00754 + 0.999i)5-s + (−0.597 + 0.434i)6-s + 0.929i·7-s + (−0.207 − 0.286i)8-s + (−0.0284 − 0.0874i)9-s + (0.223 − 0.670i)10-s + (−0.0792 + 0.243i)11-s + (0.496 − 0.161i)12-s + (0.419 − 0.136i)13-s + (0.203 − 0.624i)14-s + (0.840 + 0.620i)15-s + (0.0772 + 0.237i)16-s + (0.983 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(8.01919\)
Root analytic conductor: \(2.83181\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :5/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.38303 + 0.390517i\)
\(L(\frac12)\) \(\approx\) \(1.38303 + 0.390517i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.80 + 1.23i)T \)
5 \( 1 + (0.421 - 55.9i)T \)
good3 \( 1 + (-9.57 + 13.1i)T + (-75.0 - 231. i)T^{2} \)
7 \( 1 - 120. iT - 1.68e4T^{2} \)
11 \( 1 + (31.8 - 97.9i)T + (-1.30e5 - 9.46e4i)T^{2} \)
13 \( 1 + (-255. + 83.1i)T + (3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (-1.17e3 - 1.61e3i)T + (-4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (-666. + 484. i)T + (7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 + (-1.37e3 - 445. i)T + (5.20e6 + 3.78e6i)T^{2} \)
29 \( 1 + (5.44e3 + 3.95e3i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (-4.58e3 + 3.33e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (1.05e4 - 3.43e3i)T + (5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (-2.55e3 - 7.85e3i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 - 352. iT - 1.47e8T^{2} \)
47 \( 1 + (9.38e3 - 1.29e4i)T + (-7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (4.42e3 - 6.08e3i)T + (-1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-3.90e3 - 1.20e4i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (-1.19e4 + 3.67e4i)T + (-6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 + (2.23e4 + 3.07e4i)T + (-4.17e8 + 1.28e9i)T^{2} \)
71 \( 1 + (-4.13e3 - 3.00e3i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (-5.70e3 - 1.85e3i)T + (1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (6.10e4 + 4.43e4i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (-6.98e4 - 9.61e4i)T + (-1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + (-3.44e4 + 1.06e5i)T + (-4.51e9 - 3.28e9i)T^{2} \)
97 \( 1 + (-8.15e4 + 1.12e5i)T + (-2.65e9 - 8.16e9i)T^{2} \)
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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

−14.72769725532654082506035290779, −13.42893715156915181992656699009, −12.30136721276664722436785006669, −11.08034064971898827631316141840, −9.810393558741901484497326503208, −8.364764073403267339563799546891, −7.46781500317258561088853431882, −6.08879595836404959122963459565, −3.10483601008498527301926134586, −1.80930175427008330935626931317, 0.926493980880100675286756066343, 3.58015658239091639556443048519, 5.17045501612178604066991107280, 7.25162602071261337367398675182, 8.609051841409766682570250050534, 9.480821607156410265862414713263, 10.47425838020979933589795434722, 11.98471376617123926864559965172, 13.56517442930067946854913024554, 14.56492380378274754089961013262

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