↑

Properties

Label 2-2100-35.13-c1-0-4
Degree $2$
Conductor $2100$
Sign $-0.978 - 0.208i$
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  + (−0.707 + 0.707i)3-s + (−0.884 + 2.49i)7-s − 1.00i·9-s + 3.27·11-s + (−3.02 + 3.02i)13-s + (1.41 + 1.41i)17-s − 2.15·19-s + (−1.13 − 2.38i)21-s + (0.296 + 0.296i)23-s + (0.707 + 0.707i)27-s + 0.725i·29-s − 1.31i·31-s + (−2.31 + 2.31i)33-s + (−1.56 + 1.56i)37-s − 4.27i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.334 + 0.942i)7-s − 0.333i·9-s + 0.987·11-s + (−0.838 + 0.838i)13-s + (0.342 + 0.342i)17-s − 0.493·19-s + (−0.248 − 0.521i)21-s + (0.0617 + 0.0617i)23-s + (0.136 + 0.136i)27-s + 0.134i·29-s − 0.235i·31-s + (−0.403 + 0.403i)33-s + (−0.256 + 0.256i)37-s − 0.684i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.978 - 0.208i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.978 - 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7472824332\)
\(L(\frac12)\) \(\approx\) \(0.7472824332\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.884 - 2.49i)T \)
good11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 + (3.02 - 3.02i)T - 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + 2.15T + 19T^{2} \)
23 \( 1 + (-0.296 - 0.296i)T + 23iT^{2} \)
29 \( 1 - 0.725iT - 29T^{2} \)
31 \( 1 + 1.31iT - 31T^{2} \)
37 \( 1 + (1.56 - 1.56i)T - 37iT^{2} \)
41 \( 1 - 5.25iT - 41T^{2} \)
43 \( 1 + (0.632 + 0.632i)T + 43iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 + (3.71 + 3.71i)T + 53iT^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 - 1.31iT - 61T^{2} \)
67 \( 1 + (-3.08 + 3.08i)T - 67iT^{2} \)
71 \( 1 - 9.27T + 71T^{2} \)
73 \( 1 + (7.07 - 7.07i)T - 73iT^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 + (11.7 - 11.7i)T - 83iT^{2} \)
89 \( 1 + 18.2T + 89T^{2} \)
97 \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \)
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

−9.468330433754720294242887321335, −8.896773421027328315728588972101, −8.026680850001413414312352594683, −6.86524382642475294877887402634, −6.35780653933848301265190306918, −5.50960130825219107576139171466, −4.64630316735803631555187796007, −3.82649494748603326633887807739, −2.73892021140175452951243735071, −1.58946476737879117174633165553, 0.28360404405028755546464028015, 1.44012719567720811261460425336, 2.80118499899797799301370477302, 3.82584040489245370060071853574, 4.69352941588145169392093209117, 5.63623567767934507849771468756, 6.50046628710744093198954823288, 7.17142168355374583580602742902, 7.74612624232029558273161211290, 8.724887389814076440232008912695

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