Properties

Label 2-2100-15.2-c1-0-16
Degree $2$
Conductor $2100$
Sign $0.826 + 0.562i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.16 − 1.27i)3-s + (−0.707 − 0.707i)7-s + (−0.263 + 2.98i)9-s − 2.66i·11-s + (−1.79 + 1.79i)13-s + (1.27 − 1.27i)17-s + 5.59i·19-s + (−0.0762 + 1.73i)21-s + (6.30 + 6.30i)23-s + (4.12 − 3.15i)27-s + 0.475·29-s − 0.444·31-s + (−3.40 + 3.12i)33-s + (−3.01 − 3.01i)37-s + (4.38 + 0.193i)39-s + ⋯
L(s)  = 1  + (−0.675 − 0.737i)3-s + (−0.267 − 0.267i)7-s + (−0.0879 + 0.996i)9-s − 0.804i·11-s + (−0.497 + 0.497i)13-s + (0.308 − 0.308i)17-s + 1.28i·19-s + (−0.0166 + 0.377i)21-s + (1.31 + 1.31i)23-s + (0.794 − 0.607i)27-s + 0.0883·29-s − 0.0797·31-s + (−0.593 + 0.543i)33-s + (−0.494 − 0.494i)37-s + (0.702 + 0.0309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.826 + 0.562i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.213839650\)
\(L(\frac12)\) \(\approx\) \(1.213839650\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.16 + 1.27i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 2.66iT - 11T^{2} \)
13 \( 1 + (1.79 - 1.79i)T - 13iT^{2} \)
17 \( 1 + (-1.27 + 1.27i)T - 17iT^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 + (-6.30 - 6.30i)T + 23iT^{2} \)
29 \( 1 - 0.475T + 29T^{2} \)
31 \( 1 + 0.444T + 31T^{2} \)
37 \( 1 + (3.01 + 3.01i)T + 37iT^{2} \)
41 \( 1 + 4.01iT - 41T^{2} \)
43 \( 1 + (-6.42 + 6.42i)T - 43iT^{2} \)
47 \( 1 + (-0.964 + 0.964i)T - 47iT^{2} \)
53 \( 1 + (-0.484 - 0.484i)T + 53iT^{2} \)
59 \( 1 - 8.72T + 59T^{2} \)
61 \( 1 + 2.10T + 61T^{2} \)
67 \( 1 + (-8.71 - 8.71i)T + 67iT^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (8.99 - 8.99i)T - 73iT^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (-6.12 - 6.12i)T + 83iT^{2} \)
89 \( 1 + 5.60T + 89T^{2} \)
97 \( 1 + (-12.6 - 12.6i)T + 97iT^{2} \)
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

−8.985876467795445527421563595599, −8.120468402618477531887756404874, −7.26687744203404365900618498448, −6.86437646735966228152827998864, −5.68799615462121805916416905599, −5.42042358352185713816835227346, −4.12746514278679767976377138858, −3.13257363631174517176769994469, −1.89526288250136903137356253672, −0.75869014136730154480107050084, 0.74602531909922404613292219468, 2.49417272032559719730225347226, 3.37574528465964724506526123587, 4.68885199963925302758142172469, 4.88281248992282318467178409465, 6.01180352756877600296643996594, 6.72701824098795640181382816988, 7.48132968885351610128917170134, 8.661296501330798468044792612343, 9.246883562648285118801814996504

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