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Properties

Label 2-10e2-5.4-c9-0-10
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  + 12.2i·3-s − 9.62e3i·7-s + 1.95e4·9-s + 5.56e4·11-s + 1.69e5i·13-s − 2.07e5i·17-s − 8.02e5·19-s + 1.17e5·21-s − 1.24e6i·23-s + 4.78e5i·27-s + 4.28e6·29-s − 3.58e6·31-s + 6.79e5i·33-s + 2.89e6i·37-s − 2.07e6·39-s + ⋯
L(s)  = 1  + 0.0870i·3-s − 1.51i·7-s + 0.992·9-s + 1.14·11-s + 1.64i·13-s − 0.602i·17-s − 1.41·19-s + 0.131·21-s − 0.925i·23-s + 0.173i·27-s + 1.12·29-s − 0.697·31-s + 0.0997i·33-s + 0.254i·37-s − 0.143·39-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

\(L(5)\) \(\approx\) \(1.89847 - 1.17332i\)
\(L(\frac12)\) \(\approx\) \(1.89847 - 1.17332i\)
\(L(\frac{11}{2})\) 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 \( 1 - 12.2iT - 1.96e4T^{2} \)
7 \( 1 + 9.62e3iT - 4.03e7T^{2} \)
11 \( 1 - 5.56e4T + 2.35e9T^{2} \)
13 \( 1 - 1.69e5iT - 1.06e10T^{2} \)
17 \( 1 + 2.07e5iT - 1.18e11T^{2} \)
19 \( 1 + 8.02e5T + 3.22e11T^{2} \)
23 \( 1 + 1.24e6iT - 1.80e12T^{2} \)
29 \( 1 - 4.28e6T + 1.45e13T^{2} \)
31 \( 1 + 3.58e6T + 2.64e13T^{2} \)
37 \( 1 - 2.89e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.51e7T + 3.27e14T^{2} \)
43 \( 1 + 2.00e7iT - 5.02e14T^{2} \)
47 \( 1 + 3.73e7iT - 1.11e15T^{2} \)
53 \( 1 + 2.55e7iT - 3.29e15T^{2} \)
59 \( 1 - 9.96e7T + 8.66e15T^{2} \)
61 \( 1 - 2.00e8T + 1.16e16T^{2} \)
67 \( 1 - 8.09e7iT - 2.72e16T^{2} \)
71 \( 1 + 4.31e7T + 4.58e16T^{2} \)
73 \( 1 + 3.40e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.81e8T + 1.19e17T^{2} \)
83 \( 1 + 6.01e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.39e8T + 3.50e17T^{2} \)
97 \( 1 + 4.23e8iT - 7.60e17T^{2} \)
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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

−11.82768962501471940694230857518, −10.73145392646566748087486575821, −9.818203198518034938036399139561, −8.701684820495093666841839285238, −7.04274304739627712749868028358, −6.68140927959006282084802932387, −4.39490308523010856299416834566, −4.02361708428339548886355441616, −1.85963845756409722683320909958, −0.66808915609211207551638267930, 1.19500234071844561121516419736, 2.52054346068563088058166648823, 4.01940421334872634020396089720, 5.52110894303943646844059880741, 6.48959362326266919930338897695, 7.977651016796078644147375176366, 8.969818320210103680976055978383, 10.03895145966978557702684030692, 11.25806199041167306793626672188, 12.55613458085034279040612834756

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