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Properties

Label 2-2100-35.27-c1-0-6
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: $\chi_{2100} (1357, \cdot )$
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

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

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