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Properties

Label 2-2100-105.17-c0-0-2
Degree $2$
Conductor $2100$
Sign $0.0560 + 0.998i$
Analytic cond. $1.04803$
Root an. cond. $1.02373$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.707 − 0.707i)7-s + (0.866 − 0.499i)9-s + (−1.41 − 1.41i)13-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (−1.5 − 0.866i)31-s + (0.448 − 1.67i)37-s + (1.73 + 1.00i)39-s + (−1.22 + 1.22i)43-s − 1.00i·49-s + (1.5 − 0.866i)61-s + (0.258 − 0.965i)63-s + (0.965 − 0.258i)73-s + (0.866 − 0.5i)79-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.707 − 0.707i)7-s + (0.866 − 0.499i)9-s + (−1.41 − 1.41i)13-s + (−0.500 + 0.866i)21-s + (−0.707 + 0.707i)27-s + (−1.5 − 0.866i)31-s + (0.448 − 1.67i)37-s + (1.73 + 1.00i)39-s + (−1.22 + 1.22i)43-s − 1.00i·49-s + (1.5 − 0.866i)61-s + (0.258 − 0.965i)63-s + (0.965 − 0.258i)73-s + (0.866 − 0.5i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0560 + 0.998i$
Analytic conductor: \(1.04803\)
Root analytic conductor: \(1.02373\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :0),\ 0.0560 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6921073332\)
\(L(\frac12)\) \(\approx\) \(0.6921073332\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−9.397369009358246004137965924135, −8.071103083110663958378292077307, −7.52900494861069143278501816541, −6.84216261333754719761662618008, −5.70100850848843929816607374033, −5.17238138674655012458035596247, −4.40422622272978833381476674233, −3.45327380785973644971106557594, −2.03757011849520878481465676673, −0.55337398647452010738487467046, 1.59210099542109706442567642959, 2.39614075720857589713401689299, 3.97480310746387872597649415265, 5.02162789247560460614162068322, 5.22989982985694699514846896103, 6.45910989544835669009763139893, 7.00184892370815675734078463505, 7.79818427497818660301527016478, 8.740157875364433313411169630441, 9.519001126053961240296717318351

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