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Properties

Label 2-10e2-5.4-c9-0-6
Degree $2$
Conductor $100$
Sign $0.447 + 0.894i$
Analytic cond. $51.5035$
Root an. cond. $7.17659$
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  − 272. i·3-s + 1.00e4i·7-s − 5.44e4·9-s + 4.70e4·11-s + 9.36e3i·13-s − 1.08e5i·17-s + 6.65e5·19-s + 2.72e6·21-s + 5.76e5i·23-s + 9.45e6i·27-s + 2.61e6·29-s + 3.87e6·31-s − 1.28e7i·33-s − 1.41e7i·37-s + 2.54e6·39-s + ⋯
L(s)  = 1  − 1.94i·3-s + 1.57i·7-s − 2.76·9-s + 0.969·11-s + 0.0909i·13-s − 0.315i·17-s + 1.17·19-s + 3.05·21-s + 0.429i·23-s + 3.42i·27-s + 0.685·29-s + 0.754·31-s − 1.88i·33-s − 1.24i·37-s + 0.176·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(51.5035\)
Root analytic conductor: \(7.17659\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :9/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.74856 - 1.08067i\)
\(L(\frac12)\) \(\approx\) \(1.74856 - 1.08067i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 272. iT - 1.96e4T^{2} \)
7 \( 1 - 1.00e4iT - 4.03e7T^{2} \)
11 \( 1 - 4.70e4T + 2.35e9T^{2} \)
13 \( 1 - 9.36e3iT - 1.06e10T^{2} \)
17 \( 1 + 1.08e5iT - 1.18e11T^{2} \)
19 \( 1 - 6.65e5T + 3.22e11T^{2} \)
23 \( 1 - 5.76e5iT - 1.80e12T^{2} \)
29 \( 1 - 2.61e6T + 1.45e13T^{2} \)
31 \( 1 - 3.87e6T + 2.64e13T^{2} \)
37 \( 1 + 1.41e7iT - 1.29e14T^{2} \)
41 \( 1 - 4.62e6T + 3.27e14T^{2} \)
43 \( 1 - 8.31e6iT - 5.02e14T^{2} \)
47 \( 1 + 2.51e7iT - 1.11e15T^{2} \)
53 \( 1 - 3.49e7iT - 3.29e15T^{2} \)
59 \( 1 + 6.71e6T + 8.66e15T^{2} \)
61 \( 1 + 4.75e6T + 1.16e16T^{2} \)
67 \( 1 - 1.38e8iT - 2.72e16T^{2} \)
71 \( 1 - 3.54e8T + 4.58e16T^{2} \)
73 \( 1 - 2.41e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.61e8T + 1.19e17T^{2} \)
83 \( 1 + 6.55e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.00e9T + 3.50e17T^{2} \)
97 \( 1 + 1.24e9iT - 7.60e17T^{2} \)
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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

−11.93202186593798852530428096472, −11.53441284996936996668716377390, −9.291804207652414392383797549923, −8.469583959061583603084619033485, −7.34412420514604666718390365178, −6.30493506223379429858111601372, −5.45871591424117078036100481359, −2.97436888865915693018781726461, −1.96053991166104803637153222117, −0.846345034071271014751409264547, 0.790435777316921228349370026939, 3.22924862882152045191048659691, 4.07777571561675816803500289952, 4.92956534715695437065149202661, 6.47227501948201427336652770684, 8.098657821122497039631896594684, 9.372838485875963634265812396023, 10.11979312848051562504944346365, 10.87678431734380096637876439720, 11.84462599387143302094161526817

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