↑

Properties

Label 2-2100-35.3-c1-0-2
Degree $2$
Conductor $2100$
Sign $-0.977 - 0.212i$
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  + (0.258 + 0.965i)3-s + (−1.06 − 2.42i)7-s + (−0.866 + 0.499i)9-s + (−1.81 + 3.13i)11-s + (4.28 − 4.28i)13-s + (−5.99 + 1.60i)17-s + (−0.974 − 1.68i)19-s + (2.06 − 1.65i)21-s + (−1.48 + 5.55i)23-s + (−0.707 − 0.707i)27-s + 6.10i·29-s + (1.14 + 0.659i)31-s + (−3.49 − 0.937i)33-s + (−11.5 − 3.08i)37-s + (5.24 + 3.02i)39-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (−0.403 − 0.915i)7-s + (−0.288 + 0.166i)9-s + (−0.545 + 0.945i)11-s + (1.18 − 1.18i)13-s + (−1.45 + 0.389i)17-s + (−0.223 − 0.387i)19-s + (0.450 − 0.361i)21-s + (−0.310 + 1.15i)23-s + (−0.136 − 0.136i)27-s + 1.13i·29-s + (0.205 + 0.118i)31-s + (−0.608 − 0.163i)33-s + (−1.89 − 0.507i)37-s + (0.840 + 0.485i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.977 - 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4077604665\)
\(L(\frac12)\) \(\approx\) \(0.4077604665\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.06 + 2.42i)T \)
good11 \( 1 + (1.81 - 3.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.28 + 4.28i)T - 13iT^{2} \)
17 \( 1 + (5.99 - 1.60i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.974 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.48 - 5.55i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.10iT - 29T^{2} \)
31 \( 1 + (-1.14 - 0.659i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (11.5 + 3.08i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.50iT - 41T^{2} \)
43 \( 1 + (-6.04 - 6.04i)T + 43iT^{2} \)
47 \( 1 + (2.09 - 7.81i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.67 - 2.59i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.43 + 5.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.74 - 3.31i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.825 + 3.08i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.00T + 71T^{2} \)
73 \( 1 + (-0.767 - 2.86i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.9 - 6.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.26 - 4.26i)T - 83iT^{2} \)
89 \( 1 + (5.82 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.89 - 8.89i)T + 97iT^{2} \)
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

−9.464267986665148542370209552063, −8.773146663949688191090129286142, −7.921297106387295662624844602920, −7.18520714949471048713028225990, −6.35806032311151155604388687775, −5.40366681743938860056359433105, −4.52521989931876901320546129621, −3.74327104093033006082579509203, −2.93107853404715275175982763666, −1.55280534534465273126445252428, 0.13275985183245223256950864368, 1.82530173014400145254246344735, 2.63073406836140632765100517440, 3.67125209037019533308850379677, 4.68905450060175495414738252849, 5.86936502044915743425629538728, 6.33912707758489582450048408402, 6.99612264325403362895680696378, 8.334123890043059196753550312945, 8.577338474217159327071074369671

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