↑

Properties

Label 2-2100-7.4-c1-0-16
Degree $2$
Conductor $2100$
Sign $0.900 + 0.435i$
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  + (−0.5 + 0.866i)3-s + (1.62 + 2.09i)7-s + (−0.499 − 0.866i)9-s + (2.12 − 3.67i)11-s + 5·13-s + (2.12 − 3.67i)17-s + (−1.62 − 2.80i)19-s + (−2.62 + 0.358i)21-s + (−3 − 5.19i)23-s + 0.999·27-s − 8.48·29-s + (2 − 3.46i)31-s + (2.12 + 3.67i)33-s + (−2.5 − 4.33i)37-s + (−2.5 + 4.33i)39-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.612 + 0.790i)7-s + (−0.166 − 0.288i)9-s + (0.639 − 1.10i)11-s + 1.38·13-s + (0.514 − 0.891i)17-s + (−0.371 − 0.644i)19-s + (−0.572 + 0.0782i)21-s + (−0.625 − 1.08i)23-s + 0.192·27-s − 1.57·29-s + (0.359 − 0.622i)31-s + (0.369 + 0.639i)33-s + (−0.410 − 0.711i)37-s + (−0.400 + 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.742223565\)
\(L(\frac12)\) \(\approx\) \(1.742223565\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-1.62 - 2.09i)T \)
good11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.62 + 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 + (0.878 + 1.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.12 + 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.878 - 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.86 + 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + (6.74 - 11.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.378 + 0.655i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (8.12 + 14.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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

−8.857265501524441369490038103635, −8.599028415030842167908718285859, −7.60641183769980750311380866594, −6.40572158198811388845679526116, −5.88381163859989463383228100256, −5.13952074326151028061323926432, −4.10791413320835240916775449378, −3.34234941806207302239285696847, −2.14100546230502587804246313373, −0.71100692373952625411500613478, 1.32134429660738897706380508097, 1.80768544476821815237300552685, 3.63445347801571816036582299094, 4.09856259763820800582322716322, 5.26347645184539451982804777445, 6.09067505299997001892369289197, 6.82475283436339132607556450338, 7.66586738588319374265623023488, 8.162964715318042383389506300356, 9.110440789619862533119434475934

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