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Properties

Label 2-10e2-5.2-c8-0-6
Degree $2$
Conductor $100$
Sign $0.899 - 0.437i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.899 - 0.437i$
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: \((2,\ 100,\ (\ :4),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.44992 + 0.334061i\)
\(L(\frac12)\) \(\approx\) \(1.44992 + 0.334061i\)
\(L(5)\) 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

−11.93578665748982620053872405368, −11.18450126311993401077665007446, −10.54164817957009238523169783225, −9.145370008082584329403576800703, −8.179929479442614847586645360910, −6.23902873129341884979238000654, −5.23014076471680839664032507983, −4.56940602775866284210524550293, −2.66643038618818522762890132725, −0.64783316154214908092252421316, 0.990582043582503553331151740618, 1.67716640265380640409091452792, 4.13678787301567945388630939247, 5.38870091320527771260430362979, 6.68793692388971495171107201270, 7.43801494767829180762755713789, 8.613489861990029654750412675017, 10.66594082162401559342812030052, 11.11709446136526672591065898087, 12.04037640815929621177743161841

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