Properties

Label 2-2100-15.8-c1-0-11
Degree $2$
Conductor $2100$
Sign $0.982 + 0.183i$
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  + (−0.0811 − 1.73i)3-s + (0.707 − 0.707i)7-s + (−2.98 + 0.280i)9-s − 2.33i·11-s + (3.22 + 3.22i)13-s + (4.22 + 4.22i)17-s + 7.12i·19-s + (−1.28 − 1.16i)21-s + (−5.87 + 5.87i)23-s + (0.728 + 5.14i)27-s + 10.6·29-s + 8.24·31-s + (−4.03 + 0.189i)33-s + (−3.62 + 3.62i)37-s + (5.31 − 5.84i)39-s + ⋯
L(s)  = 1  + (−0.0468 − 0.998i)3-s + (0.267 − 0.267i)7-s + (−0.995 + 0.0935i)9-s − 0.703i·11-s + (0.894 + 0.894i)13-s + (1.02 + 1.02i)17-s + 1.63i·19-s + (−0.279 − 0.254i)21-s + (−1.22 + 1.22i)23-s + (0.140 + 0.990i)27-s + 1.97·29-s + 1.48·31-s + (−0.702 + 0.0329i)33-s + (−0.595 + 0.595i)37-s + (0.851 − 0.935i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.982 + 0.183i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.982 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.773075462\)
\(L(\frac12)\) \(\approx\) \(1.773075462\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0811 + 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 + (-3.22 - 3.22i)T + 13iT^{2} \)
17 \( 1 + (-4.22 - 4.22i)T + 17iT^{2} \)
19 \( 1 - 7.12iT - 19T^{2} \)
23 \( 1 + (5.87 - 5.87i)T - 23iT^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 8.24T + 31T^{2} \)
37 \( 1 + (3.62 - 3.62i)T - 37iT^{2} \)
41 \( 1 + 4.66iT - 41T^{2} \)
43 \( 1 + (4.41 + 4.41i)T + 43iT^{2} \)
47 \( 1 + (1.64 + 1.64i)T + 47iT^{2} \)
53 \( 1 + (-5.87 + 5.87i)T - 53iT^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + (2.03 - 2.03i)T - 67iT^{2} \)
71 \( 1 + 1.02iT - 71T^{2} \)
73 \( 1 + (-4.24 - 4.24i)T + 73iT^{2} \)
79 \( 1 - 1.43iT - 79T^{2} \)
83 \( 1 + (5.87 - 5.87i)T - 83iT^{2} \)
89 \( 1 - 9.32T + 89T^{2} \)
97 \( 1 + (0.397 - 0.397i)T - 97iT^{2} \)
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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

−8.680106512575252194549166335484, −8.246091926883696748137824910056, −7.71982735798021719983129353094, −6.54601342807017437520226267832, −6.13388786185051522886459844485, −5.34563985629166551452539292133, −3.96701800664124273469170334546, −3.29489782232168458251071187743, −1.83720186459402785456656289928, −1.14992457426787526299227422760, 0.75278761888070431327741991250, 2.59143907509758270355996273155, 3.18796246007848856847145021706, 4.58691860256994796437424677195, 4.78913160924355955104805320866, 5.90624127717835427646731478959, 6.60950646790710329676348709992, 7.81359605933882091098204121396, 8.440868821142608271210328344576, 9.134179058391126720065497329918

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