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Properties

Label 2-2100-15.2-c1-0-8
Degree $2$
Conductor $2100$
Sign $-0.872 - 0.487i$
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  + (1.27 + 1.16i)3-s + (−0.707 − 0.707i)7-s + (0.263 + 2.98i)9-s + 2.66i·11-s + (−1.79 + 1.79i)13-s + (−1.27 + 1.27i)17-s + 5.59i·19-s + (−0.0762 − 1.73i)21-s + (−6.30 − 6.30i)23-s + (−3.15 + 4.12i)27-s − 0.475·29-s − 0.444·31-s + (−3.12 + 3.40i)33-s + (−3.01 − 3.01i)37-s + (−4.38 + 0.193i)39-s + ⋯
L(s)  = 1  + (0.737 + 0.675i)3-s + (−0.267 − 0.267i)7-s + (0.0879 + 0.996i)9-s + 0.804i·11-s + (−0.497 + 0.497i)13-s + (−0.308 + 0.308i)17-s + 1.28i·19-s + (−0.0166 − 0.377i)21-s + (−1.31 − 1.31i)23-s + (−0.607 + 0.794i)27-s − 0.0883·29-s − 0.0797·31-s + (−0.543 + 0.593i)33-s + (−0.494 − 0.494i)37-s + (−0.702 + 0.0309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.872 - 0.487i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.872 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.336891745\)
\(L(\frac12)\) \(\approx\) \(1.336891745\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 1.16i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - 2.66iT - 11T^{2} \)
13 \( 1 + (1.79 - 1.79i)T - 13iT^{2} \)
17 \( 1 + (1.27 - 1.27i)T - 17iT^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 + (6.30 + 6.30i)T + 23iT^{2} \)
29 \( 1 + 0.475T + 29T^{2} \)
31 \( 1 + 0.444T + 31T^{2} \)
37 \( 1 + (3.01 + 3.01i)T + 37iT^{2} \)
41 \( 1 - 4.01iT - 41T^{2} \)
43 \( 1 + (-6.42 + 6.42i)T - 43iT^{2} \)
47 \( 1 + (0.964 - 0.964i)T - 47iT^{2} \)
53 \( 1 + (0.484 + 0.484i)T + 53iT^{2} \)
59 \( 1 + 8.72T + 59T^{2} \)
61 \( 1 + 2.10T + 61T^{2} \)
67 \( 1 + (-8.71 - 8.71i)T + 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (8.99 - 8.99i)T - 73iT^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (6.12 + 6.12i)T + 83iT^{2} \)
89 \( 1 - 5.60T + 89T^{2} \)
97 \( 1 + (-12.6 - 12.6i)T + 97iT^{2} \)
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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

−9.487373256509186905769663878127, −8.695066094423307642225645316854, −7.963895661640554168417195270030, −7.25188416841615947096279117718, −6.32931562674564994167129341142, −5.32015328944186823005214061246, −4.24163861202280484087515680540, −3.94715659003189826948596188178, −2.62621922524575910745820690383, −1.82430697533689643677020303399, 0.39290710144637313420007221687, 1.83920141635513189587547441253, 2.85729743409522154329853158841, 3.50438792775675800874155488236, 4.71563246441938647349579153746, 5.80059553075874983032450146847, 6.44492210197147588157652538591, 7.43296443878071368421899403857, 7.87847365272140069246665612746, 8.868158118121178011687456795986

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