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Properties

Label 2-10e2-5.2-c8-0-11
Degree $2$
Conductor $100$
Sign $-0.608 - 0.793i$
Analytic cond. $40.7378$
Root an. cond. $6.38262$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (90.6 − 90.6i)3-s + (−2.97e3 − 2.97e3i)7-s − 9.86e3i·9-s − 6.64e3·11-s + (−2.98e4 + 2.98e4i)13-s + (7.03e4 + 7.03e4i)17-s − 4.10e4i·19-s − 5.38e5·21-s + (−3.51e5 + 3.51e5i)23-s + (−2.99e5 − 2.99e5i)27-s − 6.59e5i·29-s + 1.00e6·31-s + (−6.02e5 + 6.02e5i)33-s + (1.35e4 + 1.35e4i)37-s + 5.41e6i·39-s + ⋯
L(s)  = 1  + (1.11 − 1.11i)3-s + (−1.23 − 1.23i)7-s − 1.50i·9-s − 0.453·11-s + (−1.04 + 1.04i)13-s + (0.842 + 0.842i)17-s − 0.314i·19-s − 2.76·21-s + (−1.25 + 1.25i)23-s + (−0.563 − 0.563i)27-s − 0.932i·29-s + 1.08·31-s + (−0.507 + 0.507i)33-s + (0.00723 + 0.00723i)37-s + 2.33i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(40.7378\)
Root analytic conductor: \(6.38262\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :4),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.201663 + 0.408807i\)
\(L(\frac12)\) \(\approx\) \(0.201663 + 0.408807i\)
\(L(5)\) 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 \( 1 + (-90.6 + 90.6i)T - 6.56e3iT^{2} \)
7 \( 1 + (2.97e3 + 2.97e3i)T + 5.76e6iT^{2} \)
11 \( 1 + 6.64e3T + 2.14e8T^{2} \)
13 \( 1 + (2.98e4 - 2.98e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-7.03e4 - 7.03e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 4.10e4iT - 1.69e10T^{2} \)
23 \( 1 + (3.51e5 - 3.51e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 6.59e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.00e6T + 8.52e11T^{2} \)
37 \( 1 + (-1.35e4 - 1.35e4i)T + 3.51e12iT^{2} \)
41 \( 1 - 1.01e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.43e6 + 2.43e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (4.91e6 + 4.91e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (8.64e6 - 8.64e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 1.29e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.00e7T + 1.91e14T^{2} \)
67 \( 1 + (1.52e7 + 1.52e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 4.11e7T + 6.45e14T^{2} \)
73 \( 1 + (1.73e7 - 1.73e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 2.57e7iT - 1.51e15T^{2} \)
83 \( 1 + (5.97e7 - 5.97e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 3.34e7iT - 3.93e15T^{2} \)
97 \( 1 + (1.14e7 + 1.14e7i)T + 7.83e15iT^{2} \)
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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

−11.96652547736845147725980873980, −10.17638077637569441811204085625, −9.428438564990390837782521212925, −7.928023194882155089909293441462, −7.28266532353304041487586940966, −6.27036237108374822515222973799, −4.04275410044438148140141621480, −2.89424752418354144190209566543, −1.58139953080654495329620612805, −0.10227697867181620138253633792, 2.68728849122081953277280267193, 3.05710799264954046496137482897, 4.72092118661008768269328452647, 5.95639657167360484848237844703, 7.77872783633453840613758175150, 8.799947980739264027551960197297, 9.810469059108087322135037263477, 10.20924166059897891938571420344, 12.11246320279936917643555542879, 12.90022018451065134713114741287

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