↑

Properties

Label 2-2100-21.17-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.744 - 0.667i$
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  + (−1.5 − 0.866i)3-s + (2.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + 5.19i·13-s + (−7.5 + 4.33i)19-s + (−3 − 3.46i)21-s − 5.19i·27-s + (−1.5 − 0.866i)31-s + (−5.5 − 9.52i)37-s + (4.5 − 7.79i)39-s − 13·43-s + (5.5 + 4.33i)49-s + 15·57-s + (−6 + 3.46i)61-s + (1.5 + 7.79i)63-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (0.944 + 0.327i)7-s + (0.5 + 0.866i)9-s + 1.44i·13-s + (−1.72 + 0.993i)19-s + (−0.654 − 0.755i)21-s − 0.999i·27-s + (−0.269 − 0.155i)31-s + (−0.904 − 1.56i)37-s + (0.720 − 1.24i)39-s − 1.98·43-s + (0.785 + 0.618i)49-s + 1.98·57-s + (−0.768 + 0.443i)61-s + (0.188 + 0.981i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5043045006\)
\(L(\frac12)\) \(\approx\) \(0.5043045006\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.5 - 4.33i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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

−9.300839802611893153596089769989, −8.527551603910247379044078972942, −7.83334304194280174316062291826, −6.93385992547203130049730781567, −6.30239586398618453138275953825, −5.48076066362850136042797694548, −4.64186524005951324616790341410, −3.93255412466942509634245344692, −2.11894608571276078040737436309, −1.63558196631155314032402485237, 0.19601286425132859264478389701, 1.53126475426790440779250902111, 2.98991685911478890352629040315, 4.09133068862827969131728565938, 4.88061314870203617498784745779, 5.41166158499978157116780263118, 6.44410892990600734520112704478, 7.09086944020698758580418432798, 8.208258444802336880104733326860, 8.640983849520001903847421880996

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