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Properties

Label 2-2100-15.8-c1-0-2
Degree $2$
Conductor $2100$
Sign $0.346 - 0.938i$
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  + (0.210 − 1.71i)3-s + (0.707 − 0.707i)7-s + (−2.91 − 0.725i)9-s + 6.43i·11-s + (−4.23 − 4.23i)13-s + (1.00 + 1.00i)17-s + 6.08i·19-s + (−1.06 − 1.36i)21-s + (−0.649 + 0.649i)23-s + (−1.86 + 4.85i)27-s + 0.486·29-s − 2.38·31-s + (11.0 + 1.35i)33-s + (−4.10 + 4.10i)37-s + (−8.17 + 6.39i)39-s + ⋯
L(s)  = 1  + (0.121 − 0.992i)3-s + (0.267 − 0.267i)7-s + (−0.970 − 0.241i)9-s + 1.93i·11-s + (−1.17 − 1.17i)13-s + (0.243 + 0.243i)17-s + 1.39i·19-s + (−0.232 − 0.297i)21-s + (−0.135 + 0.135i)23-s + (−0.358 + 0.933i)27-s + 0.0902·29-s − 0.427·31-s + (1.92 + 0.236i)33-s + (−0.674 + 0.674i)37-s + (−1.30 + 1.02i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.346 - 0.938i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9477105276\)
\(L(\frac12)\) \(\approx\) \(0.9477105276\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.210 + 1.71i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 - 6.43iT - 11T^{2} \)
13 \( 1 + (4.23 + 4.23i)T + 13iT^{2} \)
17 \( 1 + (-1.00 - 1.00i)T + 17iT^{2} \)
19 \( 1 - 6.08iT - 19T^{2} \)
23 \( 1 + (0.649 - 0.649i)T - 23iT^{2} \)
29 \( 1 - 0.486T + 29T^{2} \)
31 \( 1 + 2.38T + 31T^{2} \)
37 \( 1 + (4.10 - 4.10i)T - 37iT^{2} \)
41 \( 1 - 5.91iT - 41T^{2} \)
43 \( 1 + (-6.74 - 6.74i)T + 43iT^{2} \)
47 \( 1 + (4.70 + 4.70i)T + 47iT^{2} \)
53 \( 1 + (4.00 - 4.00i)T - 53iT^{2} \)
59 \( 1 - 8.86T + 59T^{2} \)
61 \( 1 + 7.72T + 61T^{2} \)
67 \( 1 + (8.66 - 8.66i)T - 67iT^{2} \)
71 \( 1 + 0.938iT - 71T^{2} \)
73 \( 1 + (-0.399 - 0.399i)T + 73iT^{2} \)
79 \( 1 + 6.65iT - 79T^{2} \)
83 \( 1 + (-8.35 + 8.35i)T - 83iT^{2} \)
89 \( 1 - 8.32T + 89T^{2} \)
97 \( 1 + (-1.26 + 1.26i)T - 97iT^{2} \)
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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

−9.298757003632200067839187263810, −8.093006840231711165256288921102, −7.68991564527635945851219939946, −7.15561400144462268221494321272, −6.20866862359636628162820922824, −5.30095172568591204895094609383, −4.50592292235921691130378913740, −3.26397762704286928724901818477, −2.22535527207621750259892128548, −1.38695463217357277846556988898, 0.32012583266378801599020079583, 2.26371551356318032987116510085, 3.12703620522925790736677688412, 4.04763771796536789626918463469, 4.96510292906180545245645487832, 5.55804840113677023010397289622, 6.51232690043924982630511333878, 7.47915682806546844219769306156, 8.457742206440698140506090905561, 9.073104288261139460976728559615

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