Properties

Label 2-2100-35.4-c1-0-7
Degree $2$
Conductor $2100$
Sign $-0.132 - 0.991i$
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 + (−0.358 − 2.62i)7-s + (0.499 + 0.866i)9-s + (−2.12 + 3.67i)11-s + 5.24i·13-s + (3.67 + 2.12i)17-s + (−3.5 − 6.06i)19-s + (1 − 2.44i)21-s + (−3.67 + 2.12i)23-s + 0.999i·27-s + 10.2·29-s + (−3.74 + 6.48i)31-s + (−3.67 + 2.12i)33-s + (−4.54 + 2.62i)37-s + (−2.62 + 4.54i)39-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.135 − 0.990i)7-s + (0.166 + 0.288i)9-s + (−0.639 + 1.10i)11-s + 1.45i·13-s + (0.891 + 0.514i)17-s + (−0.802 − 1.39i)19-s + (0.218 − 0.534i)21-s + (−0.766 + 0.442i)23-s + 0.192i·27-s + 1.90·29-s + (−0.672 + 1.16i)31-s + (−0.639 + 0.369i)33-s + (−0.746 + 0.430i)37-s + (−0.419 + 0.727i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.132 - 0.991i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.132 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.541524448\)
\(L(\frac12)\) \(\approx\) \(1.541524448\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.358 + 2.62i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.24iT - 13T^{2} \)
17 \( 1 + (-3.67 - 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + (3.74 - 6.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.54 - 2.62i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.34 + 4.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.878 - 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.24 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.80 + 1.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-0.655 - 0.378i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.75iT - 83T^{2} \)
89 \( 1 + (0.878 + 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.4iT - 97T^{2} \)
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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

−9.421161496937833241171290376254, −8.526080955654953834860918926512, −7.78149615120033502288603558850, −6.95936716367671486635176530628, −6.48052163288727809103486365379, −4.95642656006747071141162093459, −4.48831331424432476653960341495, −3.62271980323650479100387564274, −2.51423798632188176553402574902, −1.44828326328093578499724458956, 0.50183536063450682874778043849, 2.08790827476004179991758968553, 2.98076790308135089271614853727, 3.63187069452635646305452547692, 5.07515168627491045460374391983, 5.81896733790428361972141727224, 6.31368882162589097828612218508, 7.69451264506713632850101029767, 8.179570939463062844992881317647, 8.598886537619377762385550741237

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