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Properties

Label 2-2100-35.9-c1-0-17
Degree $2$
Conductor $2100$
Sign $0.123 + 0.992i$
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  + (−0.866 + 0.5i)3-s + (2.29 + 1.32i)7-s + (0.499 − 0.866i)9-s + (−1.82 − 3.15i)11-s − 2.64i·13-s + (3.15 − 1.82i)17-s + (−1.14 + 1.98i)19-s − 2.64·21-s + (−3.15 − 1.82i)23-s + 0.999i·27-s − 2.35·29-s + (−3.14 − 5.44i)31-s + (3.15 + 1.82i)33-s + (4.02 + 2.32i)37-s + (1.32 + 2.29i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.866 + 0.499i)7-s + (0.166 − 0.288i)9-s + (−0.549 − 0.951i)11-s − 0.733i·13-s + (0.765 − 0.442i)17-s + (−0.262 + 0.455i)19-s − 0.577·21-s + (−0.658 − 0.380i)23-s + 0.192i·27-s − 0.437·29-s + (−0.564 − 0.978i)31-s + (0.549 + 0.317i)33-s + (0.661 + 0.381i)37-s + (0.211 + 0.366i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.123 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.108613301\)
\(L(\frac12)\) \(\approx\) \(1.108613301\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.29 - 1.32i)T \)
good11 \( 1 + (1.82 + 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.64iT - 13T^{2} \)
17 \( 1 + (-3.15 + 1.82i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.14 - 1.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + (3.14 + 5.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.02 - 2.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 9.93iT - 43T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.31 + 3.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.46 + 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.02 - 2.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.35T + 71T^{2} \)
73 \( 1 + (-11.4 + 6.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.14 - 7.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.93iT - 83T^{2} \)
89 \( 1 + (-6.11 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.795266118085110126617283532803, −8.160459785956565434984789579028, −7.56140874853941652031765040044, −6.36926507191564355842506914230, −5.49736889840311820251098778237, −5.23709390289911581148082119872, −4.03119533726514579512579114288, −3.09082073565247641160760121130, −1.90208073902504460757833549772, −0.42835049425231117728612915071, 1.34531598952888915032367404814, 2.15856816219293910622319259380, 3.63208304994968894251697836001, 4.62934892763518967341765185469, 5.14710833685504871484868041065, 6.18544341196086698870638868903, 7.03848663090228346455151665292, 7.66730429562451027286246758702, 8.297894867926775251924425906812, 9.379307945373758437994710338768

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