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Properties

Label 2-2100-35.27-c1-0-8
Degree $2$
Conductor $2100$
Sign $-0.00849 - 0.999i$
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 + (0.884 + 2.49i)7-s + 1.00i·9-s + 3.27·11-s + (3.02 + 3.02i)13-s + (−1.41 + 1.41i)17-s − 2.15·19-s + (−1.13 + 2.38i)21-s + (−0.296 + 0.296i)23-s + (−0.707 + 0.707i)27-s − 0.725i·29-s + 1.31i·31-s + (2.31 + 2.31i)33-s + (1.56 + 1.56i)37-s + 4.27i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.334 + 0.942i)7-s + 0.333i·9-s + 0.987·11-s + (0.838 + 0.838i)13-s + (−0.342 + 0.342i)17-s − 0.493·19-s + (−0.248 + 0.521i)21-s + (−0.0617 + 0.0617i)23-s + (−0.136 + 0.136i)27-s − 0.134i·29-s + 0.235i·31-s + (0.403 + 0.403i)33-s + (0.256 + 0.256i)37-s + 0.684i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00849 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00849 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.00849 - 0.999i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.00849 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.177007448\)
\(L(\frac12)\) \(\approx\) \(2.177007448\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.884 - 2.49i)T \)
good11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 + (-3.02 - 3.02i)T + 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 + 2.15T + 19T^{2} \)
23 \( 1 + (0.296 - 0.296i)T - 23iT^{2} \)
29 \( 1 + 0.725iT - 29T^{2} \)
31 \( 1 - 1.31iT - 31T^{2} \)
37 \( 1 + (-1.56 - 1.56i)T + 37iT^{2} \)
41 \( 1 + 5.25iT - 41T^{2} \)
43 \( 1 + (-0.632 + 0.632i)T - 43iT^{2} \)
47 \( 1 + (1.41 - 1.41i)T - 47iT^{2} \)
53 \( 1 + (-3.71 + 3.71i)T - 53iT^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 + 1.31iT - 61T^{2} \)
67 \( 1 + (3.08 + 3.08i)T + 67iT^{2} \)
71 \( 1 - 9.27T + 71T^{2} \)
73 \( 1 + (-7.07 - 7.07i)T + 73iT^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (-11.7 - 11.7i)T + 83iT^{2} \)
89 \( 1 + 18.2T + 89T^{2} \)
97 \( 1 + (7.26 - 7.26i)T - 97iT^{2} \)
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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

−9.155952623236378638952160571292, −8.660218414630754504520957403118, −8.035593186080498162072676951913, −6.80519138404453338583552420124, −6.21914406287217633327317523284, −5.27612280100846382731095287663, −4.28696884354142153027885726838, −3.65405169836715810960452379110, −2.42397128057468323977082769096, −1.52441276010982268051680990745, 0.77156976308758973457582126619, 1.76603174913394524639755400821, 3.09575396293314880847565101514, 3.92024838161002072075460777104, 4.70261637885316849010869725690, 5.95542509758475293999182900535, 6.61255090401319471782049781983, 7.42434838332876113161125366155, 8.101094204966859697821314421921, 8.817300483222245857037831418907

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