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Properties

Label 2-2100-35.27-c1-0-15
Degree $2$
Conductor $2100$
Sign $0.989 - 0.142i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (2.63 + 0.235i)7-s + 1.00i·9-s + 4.99·11-s + (−2.63 − 2.63i)13-s + (4.14 − 4.14i)17-s + 4.66·19-s + (1.69 + 2.03i)21-s + (−4.41 + 4.41i)23-s + (−0.707 + 0.707i)27-s − 2.34i·29-s + 2.57i·31-s + (3.53 + 3.53i)33-s + (−7.06 − 7.06i)37-s − 3.72i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.996 + 0.0891i)7-s + 0.333i·9-s + 1.50·11-s + (−0.729 − 0.729i)13-s + (1.00 − 1.00i)17-s + 1.06·19-s + (0.370 + 0.443i)21-s + (−0.921 + 0.921i)23-s + (−0.136 + 0.136i)27-s − 0.436i·29-s + 0.461i·31-s + (0.614 + 0.614i)33-s + (−1.16 − 1.16i)37-s − 0.595i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.989 - 0.142i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.989 - 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.569088622\)
\(L(\frac12)\) \(\approx\) \(2.569088622\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.235i)T \)
good11 \( 1 - 4.99T + 11T^{2} \)
13 \( 1 + (2.63 + 2.63i)T + 13iT^{2} \)
17 \( 1 + (-4.14 + 4.14i)T - 17iT^{2} \)
19 \( 1 - 4.66T + 19T^{2} \)
23 \( 1 + (4.41 - 4.41i)T - 23iT^{2} \)
29 \( 1 + 2.34iT - 29T^{2} \)
31 \( 1 - 2.57iT - 31T^{2} \)
37 \( 1 + (7.06 + 7.06i)T + 37iT^{2} \)
41 \( 1 + 7.87iT - 41T^{2} \)
43 \( 1 + (-2.59 + 2.59i)T - 43iT^{2} \)
47 \( 1 + (-0.813 + 0.813i)T - 47iT^{2} \)
53 \( 1 + (-0.0317 + 0.0317i)T - 53iT^{2} \)
59 \( 1 + 0.858T + 59T^{2} \)
61 \( 1 - 1.59iT - 61T^{2} \)
67 \( 1 + (-10.5 - 10.5i)T + 67iT^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + (-7.92 - 7.92i)T + 73iT^{2} \)
79 \( 1 - 8.69iT - 79T^{2} \)
83 \( 1 + (-5.50 - 5.50i)T + 83iT^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 + (-3.51 + 3.51i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.200694348862473874807865164976, −8.389767813819083665651952300494, −7.52527564871503839434213426834, −7.11163207792730159875474590607, −5.59266546438751170486047409153, −5.26626784455504005227600705459, −4.11041684349550437686752575214, −3.42021509072612403276674293117, −2.24121471266022334973897182486, −1.08047989678845290713978186460, 1.24211734423064442324133892019, 1.92446362578728756608507294630, 3.28208575370054368936670620952, 4.16505036352849628579365812359, 4.96743687522464483650388798677, 6.10458498941487585073143175890, 6.76593525625119648785362461824, 7.68457830594052476491411231781, 8.194278130257423240133082345528, 9.052359236845721723920882924361

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