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Properties

Label 2-2100-35.13-c1-0-22
Degree $2$
Conductor $2100$
Sign $-0.315 + 0.948i$
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 + (0.235 − 2.63i)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 + (0.0891 − 0.996i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.315 + 0.948i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.076766813\)
\(L(\frac12)\) \(\approx\) \(1.076766813\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.235 + 2.63i)T \)
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

−8.819639430486635373894499690195, −8.306247355697857370096489944121, −6.97693856054231344094272817659, −6.67555031952878175267281548391, −5.75929971476926498695704767837, −4.57448791156515110537003448342, −4.12981009151463998358980143235, −3.19538543254273576570700129484, −1.65657460367332029153487633845, −0.40125443995141902425829813799, 1.55038824916900691114031138502, 2.18169715519379056272180706088, 3.78450600706276149496670481571, 4.33700879530697784477175387375, 5.63809055432508868319038903954, 6.28955363982894966332881754788, 6.67277828579660746754746298291, 7.85111631814367117124588480290, 8.793819004825338921554264407612, 9.005594537493208265151297532722

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