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Properties

Label 2-50-5.4-c9-0-5
Degree $2$
Conductor $50$
Sign $0.894 - 0.447i$
Analytic cond. $25.7517$
Root an. cond. $5.07462$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16i·2-s + 174i·3-s − 256·4-s + 2.78e3·6-s − 4.65e3i·7-s + 4.09e3i·8-s − 1.05e4·9-s + 2.89e4·11-s − 4.45e4i·12-s − 1.64e5i·13-s − 7.45e4·14-s + 6.55e4·16-s + 5.94e5i·17-s + 1.69e5i·18-s + 2.95e5·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.24i·3-s − 0.5·4-s + 0.876·6-s − 0.733i·7-s + 0.353i·8-s − 0.538·9-s + 0.597·11-s − 0.620i·12-s − 1.59i·13-s − 0.518·14-s + 0.250·16-s + 1.72i·17-s + 0.380i·18-s + 0.520·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(25.7517\)
Root analytic conductor: \(5.07462\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :9/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.82465 + 0.430741i\)
\(L(\frac12)\) \(\approx\) \(1.82465 + 0.430741i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16iT \)
5 \( 1 \)
good3 \( 1 - 174iT - 1.96e4T^{2} \)
7 \( 1 + 4.65e3iT - 4.03e7T^{2} \)
11 \( 1 - 2.89e4T + 2.35e9T^{2} \)
13 \( 1 + 1.64e5iT - 1.06e10T^{2} \)
17 \( 1 - 5.94e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.95e5T + 3.22e11T^{2} \)
23 \( 1 - 2.54e6iT - 1.80e12T^{2} \)
29 \( 1 - 3.72e6T + 1.45e13T^{2} \)
31 \( 1 - 2.33e6T + 2.64e13T^{2} \)
37 \( 1 + 1.08e7iT - 1.29e14T^{2} \)
41 \( 1 - 2.15e7T + 3.27e14T^{2} \)
43 \( 1 - 1.08e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.17e6iT - 1.11e15T^{2} \)
53 \( 1 - 9.81e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.61e7T + 8.66e15T^{2} \)
61 \( 1 + 4.39e7T + 1.16e16T^{2} \)
67 \( 1 - 8.15e7iT - 2.72e16T^{2} \)
71 \( 1 - 1.61e8T + 4.58e16T^{2} \)
73 \( 1 + 2.47e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.83e8T + 1.19e17T^{2} \)
83 \( 1 + 1.45e7iT - 1.86e17T^{2} \)
89 \( 1 + 4.70e8T + 3.50e17T^{2} \)
97 \( 1 - 1.17e8iT - 7.60e17T^{2} \)
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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

−13.63208259089429758720500353148, −12.43850776240441290540378319827, −10.92424109547119650085399219459, −10.31275631262194042080541805424, −9.317160356456233877177730574768, −7.83089663528276062315346178643, −5.65532206274870606920693094746, −4.19783692777209580522825917770, −3.32331821745383815194276435973, −1.11282376535141178738538947066, 0.796891934557313161269382118841, 2.38886327994890721359874780356, 4.67871740542370016467871817678, 6.39716814557642448131544843936, 7.03244260041193770465922104639, 8.445057283056486400140726836587, 9.503795441435983635435565942511, 11.67632425103022843162697652038, 12.36236405542627047038927028266, 13.76121509391862612632516365355

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