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Properties

Label 2-50-5.4-c7-0-3
Degree $2$
Conductor $50$
Sign $0.894 + 0.447i$
Analytic cond. $15.6192$
Root an. cond. $3.95211$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − 12i·3-s − 64·4-s − 96·6-s + 1.01e3i·7-s + 512i·8-s + 2.04e3·9-s + 1.09e3·11-s + 768i·12-s − 1.38e3i·13-s + 8.12e3·14-s + 4.09e3·16-s + 1.47e4i·17-s − 1.63e4i·18-s + 3.99e4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.256i·3-s − 0.5·4-s − 0.181·6-s + 1.11i·7-s + 0.353i·8-s + 0.934·9-s + 0.247·11-s + 0.128i·12-s − 0.174i·13-s + 0.791·14-s + 0.250·16-s + 0.725i·17-s − 0.660i·18-s + 1.33·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(15.6192\)
Root analytic conductor: \(3.95211\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :7/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.83628 - 0.433488i\)
\(L(\frac12)\) \(\approx\) \(1.83628 - 0.433488i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
5 \( 1 \)
good3 \( 1 + 12iT - 2.18e3T^{2} \)
7 \( 1 - 1.01e3iT - 8.23e5T^{2} \)
11 \( 1 - 1.09e3T + 1.94e7T^{2} \)
13 \( 1 + 1.38e3iT - 6.27e7T^{2} \)
17 \( 1 - 1.47e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.99e4T + 8.93e8T^{2} \)
23 \( 1 + 6.87e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.02e5T + 1.72e10T^{2} \)
31 \( 1 - 2.27e5T + 2.75e10T^{2} \)
37 \( 1 - 1.60e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.08e4T + 1.94e11T^{2} \)
43 \( 1 - 6.30e5iT - 2.71e11T^{2} \)
47 \( 1 - 4.72e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.49e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.64e6T + 2.48e12T^{2} \)
61 \( 1 - 8.27e5T + 3.14e12T^{2} \)
67 \( 1 + 1.26e5iT - 6.06e12T^{2} \)
71 \( 1 + 1.41e6T + 9.09e12T^{2} \)
73 \( 1 + 9.80e5iT - 1.10e13T^{2} \)
79 \( 1 - 3.56e6T + 1.92e13T^{2} \)
83 \( 1 + 5.67e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.19e7T + 4.42e13T^{2} \)
97 \( 1 - 8.68e6iT - 8.07e13T^{2} \)
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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

−13.74529141527531559691693716452, −12.54248173213163697767947799621, −11.90709785076245503053365901949, −10.42722492074235391499574316432, −9.303446044916557345214129673785, −8.019916630601646294454040205732, −6.25474988989732687867172532253, −4.61896909742189240005377350854, −2.80092841356221327532207406772, −1.24114563944784300568009509854, 0.973437532727465782508220360998, 3.73302126622654617646146735869, 5.00795433210253789579215058291, 6.82409305415962282149401926571, 7.67885388198659260097262111890, 9.401376843449351230267438844845, 10.30907769170181069412595686465, 11.85907832491203922833034389794, 13.43185229780909961740268757965, 14.04613745832716667552082867250

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