↑

Properties

Label 2-2100-105.59-c1-0-2
Degree $2$
Conductor $2100$
Sign $0.523 - 0.852i$
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  + (−0.478 − 1.66i)3-s + (−2.33 − 1.25i)7-s + (−2.54 + 1.59i)9-s + (−4.63 − 2.67i)11-s − 4.13·13-s + (0.134 + 0.0773i)17-s + (3.40 − 1.96i)19-s + (−0.971 + 4.47i)21-s + (−2.99 − 5.19i)23-s + (3.86 + 3.46i)27-s + 10.3i·29-s + (6.70 + 3.86i)31-s + (−2.23 + 9.00i)33-s + (9.07 − 5.24i)37-s + (1.97 + 6.88i)39-s + ⋯
L(s)  = 1  + (−0.276 − 0.961i)3-s + (−0.880 − 0.473i)7-s + (−0.847 + 0.531i)9-s + (−1.39 − 0.807i)11-s − 1.14·13-s + (0.0325 + 0.0187i)17-s + (0.781 − 0.450i)19-s + (−0.211 + 0.977i)21-s + (−0.625 − 1.08i)23-s + (0.744 + 0.667i)27-s + 1.91i·29-s + (1.20 + 0.694i)31-s + (−0.389 + 1.56i)33-s + (1.49 − 0.861i)37-s + (0.317 + 1.10i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.523 - 0.852i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.523 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3200948247\)
\(L(\frac12)\) \(\approx\) \(0.3200948247\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.478 + 1.66i)T \)
5 \( 1 \)
7 \( 1 + (2.33 + 1.25i)T \)
good11 \( 1 + (4.63 + 2.67i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.13T + 13T^{2} \)
17 \( 1 + (-0.134 - 0.0773i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.40 + 1.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.99 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + (-6.70 - 3.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.07 + 5.24i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.33T + 41T^{2} \)
43 \( 1 + 1.78iT - 43T^{2} \)
47 \( 1 + (-3.11 + 1.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.47 - 4.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.27 - 9.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.25 - 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.770 - 0.444i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (6.12 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.61 + 6.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.14iT - 83T^{2} \)
89 \( 1 + (8.58 + 14.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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

−9.116995302596430720545451889138, −8.324916912267846729923845437903, −7.45946980782455321489412227442, −7.07741756311557707983426358708, −6.09691020350654890325756115603, −5.43360204492529643397358080176, −4.51386100800373768698711362360, −2.95842228533442729785152141703, −2.63493591600193806761526750306, −0.924383234727928355159750197864, 0.14177540161464819779765822986, 2.39425592782601740124027848209, 3.01930151815372193423024881378, 4.20178432021381944236588019459, 4.94946649469364853054309051364, 5.72286824013982019922551478985, 6.38195460086853807920120210907, 7.68769633472174984345532664215, 8.018590834342880502239870671312, 9.501897756749946211317114104286

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