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Properties

Label 2-2100-15.8-c1-0-16
Degree $2$
Conductor $2100$
Sign $0.136 + 0.990i$
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  + (−0.625 − 1.61i)3-s + (−0.707 + 0.707i)7-s + (−2.21 + 2.01i)9-s + 1.94i·11-s + (−0.405 − 0.405i)13-s + (−0.0645 − 0.0645i)17-s + 0.513i·19-s + (1.58 + 0.700i)21-s + (2.07 − 2.07i)23-s + (4.64 + 2.32i)27-s + 1.78·29-s + 4.83·31-s + (3.14 − 1.21i)33-s + (5.59 − 5.59i)37-s + (−0.401 + 0.908i)39-s + ⋯
L(s)  = 1  + (−0.360 − 0.932i)3-s + (−0.267 + 0.267i)7-s + (−0.739 + 0.673i)9-s + 0.586i·11-s + (−0.112 − 0.112i)13-s + (−0.0156 − 0.0156i)17-s + 0.117i·19-s + (0.345 + 0.152i)21-s + (0.432 − 0.432i)23-s + (0.894 + 0.446i)27-s + 0.331·29-s + 0.868·31-s + (0.546 − 0.211i)33-s + (0.919 − 0.919i)37-s + (−0.0642 + 0.145i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.136 + 0.990i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.136 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.246502983\)
\(L(\frac12)\) \(\approx\) \(1.246502983\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.625 + 1.61i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 1.94iT - 11T^{2} \)
13 \( 1 + (0.405 + 0.405i)T + 13iT^{2} \)
17 \( 1 + (0.0645 + 0.0645i)T + 17iT^{2} \)
19 \( 1 - 0.513iT - 19T^{2} \)
23 \( 1 + (-2.07 + 2.07i)T - 23iT^{2} \)
29 \( 1 - 1.78T + 29T^{2} \)
31 \( 1 - 4.83T + 31T^{2} \)
37 \( 1 + (-5.59 + 5.59i)T - 37iT^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 + (4.95 + 4.95i)T + 43iT^{2} \)
47 \( 1 + (6.03 + 6.03i)T + 47iT^{2} \)
53 \( 1 + (2.74 - 2.74i)T - 53iT^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-0.645 + 0.645i)T - 67iT^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (-6.71 - 6.71i)T + 73iT^{2} \)
79 \( 1 + 9.75iT - 79T^{2} \)
83 \( 1 + (11.7 - 11.7i)T - 83iT^{2} \)
89 \( 1 + 2.42T + 89T^{2} \)
97 \( 1 + (-12.0 + 12.0i)T - 97iT^{2} \)
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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

−8.696626423493178969548743063636, −8.192118081336314519423238906047, −7.10392078462999261407133890124, −6.81471485092617591012502190349, −5.75952101251177140479668870546, −5.15116480483463053535121042888, −4.02471472655016330210894280682, −2.75626017940187648645602338613, −1.95627363672211916234676033805, −0.57719362012245744462162234986, 0.985307071460302299387978258058, 2.78699404247407975525109237701, 3.50172858045033077490081356523, 4.52504994173041921390413389185, 5.12276436086924758316263224291, 6.20936169741188130949428215615, 6.63654774117542365462251912561, 7.941365361381036987480953403723, 8.520346502028897129363637465145, 9.600058156493952498271593249763

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