↑

Properties

Label 2-2100-105.89-c1-0-14
Degree $2$
Conductor $2100$
Sign $0.532 - 0.846i$
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  + (−1.70 − 0.317i)3-s + (1.99 + 1.73i)7-s + (2.79 + 1.07i)9-s + (−3.38 + 1.95i)11-s + 6.06·13-s + (2.65 − 1.53i)17-s + (2.94 + 1.70i)19-s + (−2.85 − 3.58i)21-s + (−1.43 + 2.48i)23-s + (−4.42 − 2.72i)27-s − 7.97i·29-s + (−5.63 + 3.25i)31-s + (6.37 − 2.25i)33-s + (0.113 + 0.0654i)37-s + (−10.3 − 1.92i)39-s + ⋯
L(s)  = 1  + (−0.983 − 0.183i)3-s + (0.755 + 0.655i)7-s + (0.932 + 0.359i)9-s + (−1.01 + 0.588i)11-s + 1.68·13-s + (0.643 − 0.371i)17-s + (0.676 + 0.390i)19-s + (−0.622 − 0.782i)21-s + (−0.299 + 0.518i)23-s + (−0.851 − 0.524i)27-s − 1.48i·29-s + (−1.01 + 0.583i)31-s + (1.10 − 0.391i)33-s + (0.0186 + 0.0107i)37-s + (−1.65 − 0.307i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.532 - 0.846i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.335917988\)
\(L(\frac12)\) \(\approx\) \(1.335917988\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.317i)T \)
5 \( 1 \)
7 \( 1 + (-1.99 - 1.73i)T \)
good11 \( 1 + (3.38 - 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
17 \( 1 + (-2.65 + 1.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.94 - 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.43 - 2.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.97iT - 29T^{2} \)
31 \( 1 + (5.63 - 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.113 - 0.0654i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 - 4.43iT - 43T^{2} \)
47 \( 1 + (-8.71 - 5.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.67 - 4.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.28 + 2.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.8 + 7.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.63iT - 71T^{2} \)
73 \( 1 + (3.88 + 6.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.22 + 2.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (4.11 - 7.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.74T + 97T^{2} \)
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

−9.319019596179771416951729765247, −8.200797706579196849546134419644, −7.77445738618986592267052037371, −6.84594942683483900698548075266, −5.75062710268830288705416306105, −5.51027747460624141572915132252, −4.57972413450206664252869644463, −3.51645640406860180452056554532, −2.12403905019048848412899256583, −1.14453848330402566103238044823, 0.64818604523512130754671091803, 1.67755757250658693529980830945, 3.40743780816325612625116134710, 4.05162103592044731710168159419, 5.35072603897336370287117172055, 5.45040648396157791078747943247, 6.63006640931117836033412852476, 7.32163668182444704598430893151, 8.247575114259331200417023772462, 8.824053175908357400610441763659

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