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Properties

Label 2-2100-5.4-c1-0-16
Degree $2$
Conductor $2100$
Sign $0.447 + 0.894i$
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  + i·3-s − i·7-s − 9-s + 6·11-s − 4i·13-s − 6i·17-s − 2·19-s + 21-s − i·27-s − 6·29-s − 10·31-s + 6i·33-s − 2i·37-s + 4·39-s − 6·41-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 1.80·11-s − 1.10i·13-s − 1.45i·17-s − 0.458·19-s + 0.218·21-s − 0.192i·27-s − 1.11·29-s − 1.79·31-s + 1.04i·33-s − 0.328i·37-s + 0.640·39-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526151880\)
\(L(\frac12)\) \(\approx\) \(1.526151880\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−9.155355237928304641317401512772, −8.324123597042410467535872598423, −7.29533866146785588398474346426, −6.72968007592384811071474199259, −5.65456109008119805397148376645, −4.97811158927648025340578562254, −3.85423576287053218733809732813, −3.42843394096223263262900199321, −1.99769552241843360414095510257, −0.54789257247745583718774084444, 1.45130304828502986774155297989, 2.05370804337113063915514180621, 3.63597240633513617544941809826, 4.13010747690806675939735742786, 5.42627908550052787580229016327, 6.34479701767229900640850277198, 6.70070818044907245779431159408, 7.64120898165077936271498657367, 8.646693035088994163307033674817, 9.056905264667100048224988577378

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