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Properties

Label 2-2100-35.4-c1-0-0
Degree $2$
Conductor $2100$
Sign $-0.981 - 0.192i$
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 + (2.59 + 0.5i)7-s + (0.499 + 0.866i)9-s + (−1 + 1.73i)11-s + 3i·13-s + (−6.92 − 4i)17-s + (−0.5 − 0.866i)19-s + (−2 − 1.73i)21-s + (−6.92 + 4i)23-s − 0.999i·27-s − 4·29-s + (−1.5 + 2.59i)31-s + (1.73 − 0.999i)33-s + (−0.866 + 0.5i)37-s + (1.5 − 2.59i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.981 + 0.188i)7-s + (0.166 + 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.832i·13-s + (−1.68 − 0.970i)17-s + (−0.114 − 0.198i)19-s + (−0.436 − 0.377i)21-s + (−1.44 + 0.834i)23-s − 0.192i·27-s − 0.742·29-s + (−0.269 + 0.466i)31-s + (0.301 − 0.174i)33-s + (−0.142 + 0.0821i)37-s + (0.240 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.981 - 0.192i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1346776390\)
\(L(\frac12)\) \(\approx\) \(0.1346776390\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 0.5i)T \)
good11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + (6.92 + 4i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.92 - 4i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 + 6.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (9.52 + 5.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.285328462389812317931416657346, −8.809369362461288823160319182744, −7.71276848329371586669811889416, −7.21989952787897380941357344732, −6.36895459891869123189581787161, −5.42312728809628678368502249545, −4.70812297001818277695606852138, −3.98798267057446369007703217281, −2.33406387912292707048828490779, −1.73319764035256811439336747848, 0.04763220220394026549337265454, 1.58792058366337453654922312744, 2.72606859791110789057607943852, 4.16463188524636318822787434209, 4.47245309744062538984288304523, 5.76545344505378846726760405382, 6.05165439544645034912688440403, 7.25237600685563057335394567383, 8.098356885463762219029649549428, 8.565431810338974311138092693615

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