↑

Properties

Label 2-10e2-4.3-c10-0-81
Degree $2$
Conductor $100$
Sign $-0.748 + 0.663i$
Analytic cond. $63.5357$
Root an. cond. $7.97093$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (29.9 − 11.3i)2-s − 137. i·3-s + (765. − 679. i)4-s + (−1.55e3 − 4.10e3i)6-s − 2.49e4i·7-s + (1.51e4 − 2.90e4i)8-s + 4.02e4·9-s − 1.13e4i·11-s + (−9.32e4 − 1.05e5i)12-s + 4.07e5·13-s + (−2.82e5 − 7.45e5i)14-s + (1.24e5 − 1.04e6i)16-s + 1.49e6·17-s + (1.20e6 − 4.57e5i)18-s − 2.09e6i·19-s + ⋯
L(s)  = 1  + (0.934 − 0.354i)2-s − 0.564i·3-s + (0.748 − 0.663i)4-s + (−0.200 − 0.527i)6-s − 1.48i·7-s + (0.463 − 0.885i)8-s + 0.681·9-s − 0.0702i·11-s + (−0.374 − 0.422i)12-s + 1.09·13-s + (−0.526 − 1.38i)14-s + (0.119 − 0.992i)16-s + 1.05·17-s + (0.637 − 0.241i)18-s − 0.846i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.748 + 0.663i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ -0.748 + 0.663i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.62682 - 4.28478i\)
\(L(\frac12)\) \(\approx\) \(1.62682 - 4.28478i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-29.9 + 11.3i)T \)
5 \( 1 \)
good3 \( 1 + 137. iT - 5.90e4T^{2} \)
7 \( 1 + 2.49e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.13e4iT - 2.59e10T^{2} \)
13 \( 1 - 4.07e5T + 1.37e11T^{2} \)
17 \( 1 - 1.49e6T + 2.01e12T^{2} \)
19 \( 1 + 2.09e6iT - 6.13e12T^{2} \)
23 \( 1 - 1.00e7iT - 4.14e13T^{2} \)
29 \( 1 + 2.68e7T + 4.20e14T^{2} \)
31 \( 1 - 9.47e6iT - 8.19e14T^{2} \)
37 \( 1 - 6.03e7T + 4.80e15T^{2} \)
41 \( 1 + 3.12e7T + 1.34e16T^{2} \)
43 \( 1 + 2.55e8iT - 2.16e16T^{2} \)
47 \( 1 - 3.78e8iT - 5.25e16T^{2} \)
53 \( 1 + 7.29e6T + 1.74e17T^{2} \)
59 \( 1 + 2.59e8iT - 5.11e17T^{2} \)
61 \( 1 + 3.30e8T + 7.13e17T^{2} \)
67 \( 1 + 7.94e7iT - 1.82e18T^{2} \)
71 \( 1 - 1.16e9iT - 3.25e18T^{2} \)
73 \( 1 + 3.86e9T + 4.29e18T^{2} \)
79 \( 1 - 3.82e9iT - 9.46e18T^{2} \)
83 \( 1 - 9.01e8iT - 1.55e19T^{2} \)
89 \( 1 + 4.02e9T + 3.11e19T^{2} \)
97 \( 1 - 1.37e10T + 7.37e19T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.51427748693756452175102138923, −10.67857813822087277334996430774, −9.654770360118482747632328636874, −7.62971216832664605836971501820, −6.98668919029078211017339926339, −5.71158357746993339637201946620, −4.23836967416219886254479152151, −3.38737976948409711053316941518, −1.56517371976408911050701669109, −0.856629555057288619368377076417, 1.73948593693483507608482112845, 3.11910530926458672042267355564, 4.25096452007481915029929576124, 5.48244837032191070844293029918, 6.28477135081657396457739733008, 7.83564038781914058386037569809, 8.916894664997912343965064101665, 10.25941348364493923554450262262, 11.47887497031641871950850997222, 12.41777608635687204164173259096

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