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Properties

Label 2-2100-35.13-c1-0-18
Degree $2$
Conductor $2100$
Sign $0.525 + 0.850i$
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 + (1.87 − 1.87i)7-s − 1.00i·9-s − 2·11-s + (1.41 − 1.41i)13-s + (2.82 + 2.82i)17-s + 5.29·19-s − 2.64i·21-s + (3.74 + 3.74i)23-s + (−0.707 − 0.707i)27-s + 6i·29-s − 5.29i·31-s + (−1.41 + 1.41i)33-s − 2.00i·39-s − 10.5i·41-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.707 − 0.707i)7-s − 0.333i·9-s − 0.603·11-s + (0.392 − 0.392i)13-s + (0.685 + 0.685i)17-s + 1.21·19-s − 0.577i·21-s + (0.780 + 0.780i)23-s + (−0.136 − 0.136i)27-s + 1.11i·29-s − 0.950i·31-s + (−0.246 + 0.246i)33-s − 0.320i·39-s − 1.65i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.525 + 0.850i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.296442977\)
\(L(\frac12)\) \(\approx\) \(2.296442977\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 + 1.87i)T \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + (-3.74 - 3.74i)T + 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 + (3.74 + 3.74i)T + 43iT^{2} \)
47 \( 1 + (5.65 + 5.65i)T + 47iT^{2} \)
53 \( 1 + (-3.74 - 3.74i)T + 53iT^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 10.5iT - 61T^{2} \)
67 \( 1 + (-3.74 + 3.74i)T - 67iT^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (9.89 - 9.89i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1.41 + 1.41i)T + 97iT^{2} \)
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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

−8.784247806134647329900435515431, −8.137933813061889261779945269925, −7.45024127862643770801544746383, −6.94616202990129869544185941038, −5.62324322723384056776789136335, −5.14685034442183463338104192946, −3.83932049621438757278471179434, −3.20885513060841128181869111376, −1.88790313208900823872191124129, −0.898852267629923584680717301297, 1.26681413147620012855145036306, 2.57971626553247789576129044435, 3.23190168923105197902637353684, 4.53853007875104363797307718639, 5.10133124247218910439053137375, 5.91726464931583993532304529230, 7.01369014704845680456873816266, 7.891985552174556012242895786039, 8.414409411481681131796217036767, 9.237245020995506652505418244833

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