Properties

Label 2-2100-35.12-c1-0-13
Degree $2$
Conductor $2100$
Sign $0.279 + 0.960i$
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  + (0.258 − 0.965i)3-s + (−0.895 + 2.48i)7-s + (−0.866 − 0.499i)9-s + (−0.720 − 1.24i)11-s + (−1.97 − 1.97i)13-s + (3.65 + 0.980i)17-s + (−0.643 + 1.11i)19-s + (2.17 + 1.50i)21-s + (0.135 + 0.503i)23-s + (−0.707 + 0.707i)27-s − 6.13i·29-s + (7.63 − 4.40i)31-s + (−1.39 + 0.373i)33-s + (4.08 − 1.09i)37-s + (−2.41 + 1.39i)39-s + ⋯
L(s)  = 1  + (0.149 − 0.557i)3-s + (−0.338 + 0.940i)7-s + (−0.288 − 0.166i)9-s + (−0.217 − 0.376i)11-s + (−0.546 − 0.546i)13-s + (0.887 + 0.237i)17-s + (−0.147 + 0.255i)19-s + (0.474 + 0.329i)21-s + (0.0281 + 0.105i)23-s + (−0.136 + 0.136i)27-s − 1.13i·29-s + (1.37 − 0.791i)31-s + (−0.242 + 0.0649i)33-s + (0.671 − 0.179i)37-s + (−0.386 + 0.223i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.505346665\)
\(L(\frac12)\) \(\approx\) \(1.505346665\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.895 - 2.48i)T \)
good11 \( 1 + (0.720 + 1.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.97 + 1.97i)T + 13iT^{2} \)
17 \( 1 + (-3.65 - 0.980i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.643 - 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.135 - 0.503i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.13iT - 29T^{2} \)
31 \( 1 + (-7.63 + 4.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.08 + 1.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 9.57iT - 41T^{2} \)
43 \( 1 + (-1.27 + 1.27i)T - 43iT^{2} \)
47 \( 1 + (0.746 + 2.78i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.83 - 1.56i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.77 + 3.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.71 + 0.987i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.583 + 2.17i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.24T + 71T^{2} \)
73 \( 1 + (-0.920 + 3.43i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.09 + 4.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \)
89 \( 1 + (-3.94 + 6.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.35 - 9.35i)T - 97iT^{2} \)
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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

−8.855558405281134413797560670021, −8.065724164605673056453634144318, −7.60520222640169564152934984467, −6.46264212554111149933314666081, −5.85040407633717526285654433725, −5.16700131137073227260797045476, −3.87048123457023585636062300396, −2.85583054272778217304661698363, −2.14413269647799062960927325966, −0.59166266378083043826121832326, 1.12884413191257150331054931169, 2.64709043469955607743745179778, 3.47317866340239200180428376576, 4.50142109147096458590325424036, 4.97715800394240556030958281752, 6.19746771642095627551856427349, 6.98959405508973863823195591887, 7.67920305114078016540664058454, 8.512824937321308068165089041968, 9.474493782974146727786249030076

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