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Properties

Label 2-50-5.4-c5-0-1
Degree $2$
Conductor $50$
Sign $-0.894 + 0.447i$
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  + 4i·2-s + 24i·3-s − 16·4-s − 96·6-s + 172i·7-s − 64i·8-s − 333·9-s + 132·11-s − 384i·12-s − 946i·13-s − 688·14-s + 256·16-s + 222i·17-s − 1.33e3i·18-s − 500·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.53i·3-s − 0.5·4-s − 1.08·6-s + 1.32i·7-s − 0.353i·8-s − 1.37·9-s + 0.328·11-s − 0.769i·12-s − 1.55i·13-s − 0.938·14-s + 0.250·16-s + 0.186i·17-s − 0.968i·18-s − 0.317·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(8.01919\)
Root analytic conductor: \(2.83181\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :5/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.288707 - 1.22298i\)
\(L(\frac12)\) \(\approx\) \(0.288707 - 1.22298i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 \)
good3 \( 1 - 24iT - 243T^{2} \)
7 \( 1 - 172iT - 1.68e4T^{2} \)
11 \( 1 - 132T + 1.61e5T^{2} \)
13 \( 1 + 946iT - 3.71e5T^{2} \)
17 \( 1 - 222iT - 1.41e6T^{2} \)
19 \( 1 + 500T + 2.47e6T^{2} \)
23 \( 1 - 3.56e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.19e3T + 2.05e7T^{2} \)
31 \( 1 - 2.31e3T + 2.86e7T^{2} \)
37 \( 1 - 1.12e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.24e3T + 1.15e8T^{2} \)
43 \( 1 - 2.06e4iT - 1.47e8T^{2} \)
47 \( 1 + 6.58e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.10e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.98e3T + 7.14e8T^{2} \)
61 \( 1 - 1.66e4T + 8.44e8T^{2} \)
67 \( 1 + 1.80e3iT - 1.35e9T^{2} \)
71 \( 1 + 2.45e4T + 1.80e9T^{2} \)
73 \( 1 - 2.04e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.62e4T + 3.07e9T^{2} \)
83 \( 1 + 5.15e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.10e5T + 5.58e9T^{2} \)
97 \( 1 - 7.83e4iT - 8.58e9T^{2} \)
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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

−15.29336092450261457431818363764, −14.74007862147340381606546460093, −13.04933949864267592408124133679, −11.57119953584087347814121024176, −10.15715129845375007546848551556, −9.222746294761790308555757696222, −8.137480371239540080009968751389, −5.92070690080374976791750645134, −4.98226882760746020899683275994, −3.30404333279361613048059582740, 0.65995753499962463466220845142, 2.03109563107077180518386289464, 4.17454835899498281186338390751, 6.52847171616421580780037638759, 7.46945555023206371357806090670, 8.951442244599238116260560224313, 10.59838285220001490747616317181, 11.74873631531908036099115843733, 12.70453476788414770455840909202, 13.76046083334954565954303117818

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