Properties

Label 2-2100-15.14-c2-0-49
Degree $2$
Conductor $2100$
Sign $0.919 + 0.393i$
Analytic cond. $57.2208$
Root an. cond. $7.56444$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.99 − 0.177i)3-s − 2.64i·7-s + (8.93 − 1.06i)9-s + 14.1i·11-s − 20.2i·13-s + 21.8·17-s − 6.93·19-s + (−0.468 − 7.92i)21-s + 7.73·23-s + (26.5 − 4.76i)27-s + 37.3i·29-s + 15.2·31-s + (2.49 + 42.2i)33-s − 12.1i·37-s + (−3.58 − 60.5i)39-s + ⋯
L(s)  = 1  + (0.998 − 0.0590i)3-s − 0.377i·7-s + (0.993 − 0.117i)9-s + 1.28i·11-s − 1.55i·13-s + 1.28·17-s − 0.365·19-s + (−0.0223 − 0.377i)21-s + 0.336·23-s + (0.984 − 0.176i)27-s + 1.28i·29-s + 0.493·31-s + (0.0756 + 1.27i)33-s − 0.327i·37-s + (−0.0918 − 1.55i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(57.2208\)
Root analytic conductor: \(7.56444\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.339129195\)
\(L(\frac12)\) \(\approx\) \(3.339129195\)
\(L(2)\) 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 \( 1 + (-2.99 + 0.177i)T \)
5 \( 1 \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 14.1iT - 121T^{2} \)
13 \( 1 + 20.2iT - 169T^{2} \)
17 \( 1 - 21.8T + 289T^{2} \)
19 \( 1 + 6.93T + 361T^{2} \)
23 \( 1 - 7.73T + 529T^{2} \)
29 \( 1 - 37.3iT - 841T^{2} \)
31 \( 1 - 15.2T + 961T^{2} \)
37 \( 1 + 12.1iT - 1.36e3T^{2} \)
41 \( 1 + 42.3iT - 1.68e3T^{2} \)
43 \( 1 + 16.1iT - 1.84e3T^{2} \)
47 \( 1 + 71.8T + 2.20e3T^{2} \)
53 \( 1 - 101.T + 2.80e3T^{2} \)
59 \( 1 - 25.7iT - 3.48e3T^{2} \)
61 \( 1 - 78.1T + 3.72e3T^{2} \)
67 \( 1 + 123. iT - 4.48e3T^{2} \)
71 \( 1 + 34.5iT - 5.04e3T^{2} \)
73 \( 1 - 100. iT - 5.32e3T^{2} \)
79 \( 1 - 42.5T + 6.24e3T^{2} \)
83 \( 1 + 82.1T + 6.88e3T^{2} \)
89 \( 1 - 42.3iT - 7.92e3T^{2} \)
97 \( 1 - 48.3iT - 9.40e3T^{2} \)
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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

−8.753337486193949348125878020256, −8.087639469888316357950388876536, −7.38019203503466465055355732002, −6.88597993557548993597350619969, −5.56484210923601383849828115335, −4.79175293953436707604701922256, −3.74114722981802893696716722713, −3.05759832944366241507524085984, −2.00124897567371725984427132795, −0.880424873785458372253829944710, 1.05804624305588026836638600837, 2.19175724342642136059079700721, 3.11430823865902550378882553353, 3.90795093871432818268320780993, 4.80091089386410270787469132720, 5.92484583337563852302075476682, 6.64904757259919205454080641918, 7.58694286708902028538074005703, 8.412592692775684832952287657101, 8.768619025091758777032023696418

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