Properties

Label 2-50-25.9-c5-0-5
Degree $2$
Conductor $50$
Sign $0.996 - 0.0811i$
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  + (−2.35 + 3.23i)2-s + (−19.7 + 6.41i)3-s + (−4.94 − 15.2i)4-s + (−52.1 − 20.0i)5-s + (25.6 − 78.9i)6-s + 5.87i·7-s + (60.8 + 19.7i)8-s + (151. − 110. i)9-s + (187. − 121. i)10-s + (−9.61 − 6.98i)11-s + (195. + 268. i)12-s + (365. + 503. i)13-s + (−19.0 − 13.8i)14-s + (1.15e3 + 62.1i)15-s + (−207. + 150. i)16-s + (773. + 251. i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−1.26 + 0.411i)3-s + (−0.154 − 0.475i)4-s + (−0.933 − 0.359i)5-s + (0.290 − 0.895i)6-s + 0.0453i·7-s + (0.336 + 0.109i)8-s + (0.625 − 0.454i)9-s + (0.593 − 0.384i)10-s + (−0.0239 − 0.0174i)11-s + (0.391 + 0.538i)12-s + (0.600 + 0.826i)13-s + (−0.0259 − 0.0188i)14-s + (1.32 + 0.0713i)15-s + (−0.202 + 0.146i)16-s + (0.649 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0811i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.996 - 0.0811i$
Analytic conductor: \(8.01919\)
Root analytic conductor: \(2.83181\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :5/2),\ 0.996 - 0.0811i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.573256 + 0.0232949i\)
\(L(\frac12)\) \(\approx\) \(0.573256 + 0.0232949i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.35 - 3.23i)T \)
5 \( 1 + (52.1 + 20.0i)T \)
good3 \( 1 + (19.7 - 6.41i)T + (196. - 142. i)T^{2} \)
7 \( 1 - 5.87iT - 1.68e4T^{2} \)
11 \( 1 + (9.61 + 6.98i)T + (4.97e4 + 1.53e5i)T^{2} \)
13 \( 1 + (-365. - 503. i)T + (-1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (-773. - 251. i)T + (1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (-721. + 2.22e3i)T + (-2.00e6 - 1.45e6i)T^{2} \)
23 \( 1 + (359. - 495. i)T + (-1.98e6 - 6.12e6i)T^{2} \)
29 \( 1 + (2.35e3 + 7.25e3i)T + (-1.65e7 + 1.20e7i)T^{2} \)
31 \( 1 + (-522. + 1.60e3i)T + (-2.31e7 - 1.68e7i)T^{2} \)
37 \( 1 + (-3.76e3 - 5.18e3i)T + (-2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (3.25e3 - 2.36e3i)T + (3.58e7 - 1.10e8i)T^{2} \)
43 \( 1 + 8.27e3iT - 1.47e8T^{2} \)
47 \( 1 + (2.38e4 - 7.75e3i)T + (1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (-3.33e4 + 1.08e4i)T + (3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (-2.24e4 + 1.63e4i)T + (2.20e8 - 6.79e8i)T^{2} \)
61 \( 1 + (-4.04e4 - 2.93e4i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 + (-1.15e4 - 3.75e3i)T + (1.09e9 + 7.93e8i)T^{2} \)
71 \( 1 + (-3.39e3 - 1.04e4i)T + (-1.45e9 + 1.06e9i)T^{2} \)
73 \( 1 + (-1.82e4 + 2.51e4i)T + (-6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (4.24e3 + 1.30e4i)T + (-2.48e9 + 1.80e9i)T^{2} \)
83 \( 1 + (7.68e4 + 2.49e4i)T + (3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + (8.19e3 + 5.95e3i)T + (1.72e9 + 5.31e9i)T^{2} \)
97 \( 1 + (1.39e4 - 4.54e3i)T + (6.94e9 - 5.04e9i)T^{2} \)
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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

−15.07338860398154981861755738766, −13.40257279073363487660184209736, −11.76704035154829846166263021841, −11.24642835110684836656282246982, −9.788896343587383494274759057258, −8.378265831187070251581272031885, −6.92630084871214389280643562592, −5.55134862271001444183550589737, −4.28388658644632664136356424299, −0.57784958561652310587055206693, 0.926949921878030356401606298462, 3.52988307040083929291695089862, 5.50885084107679439557916846301, 7.08105956545431590913901789510, 8.284944046881649764738664512500, 10.23576980657275459831234455262, 11.12306547501900791587552181140, 12.02200470112571038477796440459, 12.78776811116090842997680852004, 14.54205083444930363896171553311

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