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Properties

Label 2-2100-21.17-c1-0-46
Degree $2$
Conductor $2100$
Sign $-0.515 + 0.856i$
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  + (0.990 − 1.42i)3-s + (2.64 + 0.156i)7-s + (−1.03 − 2.81i)9-s + (−4.92 − 2.84i)11-s + 1.43i·13-s + (1.62 − 2.81i)17-s + (5.43 − 3.13i)19-s + (2.83 − 3.59i)21-s + (−0.884 + 0.510i)23-s + (−5.02 − 1.31i)27-s + 4.95i·29-s + (−2.28 − 1.31i)31-s + (−8.91 + 4.17i)33-s + (−4.55 − 7.88i)37-s + (2.04 + 1.42i)39-s + ⋯
L(s)  = 1  + (0.572 − 0.820i)3-s + (0.998 + 0.0592i)7-s + (−0.345 − 0.938i)9-s + (−1.48 − 0.857i)11-s + 0.398i·13-s + (0.393 − 0.682i)17-s + (1.24 − 0.719i)19-s + (0.619 − 0.784i)21-s + (−0.184 + 0.106i)23-s + (−0.967 − 0.253i)27-s + 0.920i·29-s + (−0.410 − 0.236i)31-s + (−1.55 + 0.727i)33-s + (−0.748 − 1.29i)37-s + (0.326 + 0.227i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.932204511\)
\(L(\frac12)\) \(\approx\) \(1.932204511\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.990 + 1.42i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.156i)T \)
good11 \( 1 + (4.92 + 2.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.43iT - 13T^{2} \)
17 \( 1 + (-1.62 + 2.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.43 + 3.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.884 - 0.510i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.95iT - 29T^{2} \)
31 \( 1 + (2.28 + 1.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.203T + 41T^{2} \)
43 \( 1 - 3.91T + 43T^{2} \)
47 \( 1 + (5.76 + 9.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.21 + 5.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.739 - 1.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.62 + 4.40i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.34 - 4.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.08iT - 71T^{2} \)
73 \( 1 + (-7.82 - 4.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.80 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (-7.53 - 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−8.741114189318367825932270854662, −7.912218504466696923949433639552, −7.52764504047497253661567425725, −6.71132147367034367762926992120, −5.42359371354280663374082638488, −5.15799982943992540802192586471, −3.63943113595143475077128068620, −2.80133504436196237074507586403, −1.88382587281802677246406961678, −0.61615990160134194830465873973, 1.62738967340307133392922856176, 2.65866384991501820901725175513, 3.57637107120943145902358846914, 4.66547614450467788498058554753, 5.12132075385658904425490070033, 5.95627787334838021093288571160, 7.54502375291674798079462135065, 7.82428402965343255132248157018, 8.425533822679390030239003214504, 9.511504285525816250894160778460

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