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Properties

Label 2-2100-15.8-c1-0-21
Degree $2$
Conductor $2100$
Sign $0.824 + 0.565i$
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  + (1.15 − 1.28i)3-s + (−0.707 + 0.707i)7-s + (−0.317 − 2.98i)9-s + 1.81i·11-s + (2.28 + 2.28i)13-s + (0.614 + 0.614i)17-s + 0.915i·19-s + (0.0918 + 1.72i)21-s + (3.11 − 3.11i)23-s + (−4.21 − 3.04i)27-s + 6.83·29-s + 8.15·31-s + (2.33 + 2.10i)33-s + (1.57 − 1.57i)37-s + (5.59 − 0.297i)39-s + ⋯
L(s)  = 1  + (0.668 − 0.743i)3-s + (−0.267 + 0.267i)7-s + (−0.105 − 0.994i)9-s + 0.547i·11-s + (0.634 + 0.634i)13-s + (0.148 + 0.148i)17-s + 0.210i·19-s + (0.0200 + 0.377i)21-s + (0.649 − 0.649i)23-s + (−0.810 − 0.586i)27-s + 1.26·29-s + 1.46·31-s + (0.407 + 0.366i)33-s + (0.259 − 0.259i)37-s + (0.895 − 0.0475i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.824 + 0.565i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.824 + 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.308708429\)
\(L(\frac12)\) \(\approx\) \(2.308708429\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 1.28i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 1.81iT - 11T^{2} \)
13 \( 1 + (-2.28 - 2.28i)T + 13iT^{2} \)
17 \( 1 + (-0.614 - 0.614i)T + 17iT^{2} \)
19 \( 1 - 0.915iT - 19T^{2} \)
23 \( 1 + (-3.11 + 3.11i)T - 23iT^{2} \)
29 \( 1 - 6.83T + 29T^{2} \)
31 \( 1 - 8.15T + 31T^{2} \)
37 \( 1 + (-1.57 + 1.57i)T - 37iT^{2} \)
41 \( 1 - 0.926iT - 41T^{2} \)
43 \( 1 + (-3.34 - 3.34i)T + 43iT^{2} \)
47 \( 1 + (7.74 + 7.74i)T + 47iT^{2} \)
53 \( 1 + (-3.13 + 3.13i)T - 53iT^{2} \)
59 \( 1 - 9.50T + 59T^{2} \)
61 \( 1 - 0.778T + 61T^{2} \)
67 \( 1 + (3.95 - 3.95i)T - 67iT^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + (3.51 + 3.51i)T + 73iT^{2} \)
79 \( 1 - 8.66iT - 79T^{2} \)
83 \( 1 + (-9.57 + 9.57i)T - 83iT^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + (3.33 - 3.33i)T - 97iT^{2} \)
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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

−8.807469549250181540413722976786, −8.376941054074353534545329318320, −7.50456074011444247267411154807, −6.58568524971580534935509689814, −6.27581709189315977033038923377, −4.95859414248314016367748882540, −3.99478173430857750778920583694, −3.00376048071083925880161275758, −2.14479272391855273013110254800, −0.991183984723611650147672347698, 1.04553845026055641404651937166, 2.68686118008275698073952948987, 3.27810167506300936608138865822, 4.20949749714210807349600858006, 5.06340294638450210710822058432, 5.94673652047073416326086930103, 6.87696320500899265911915689566, 7.901727574147317133923684694740, 8.411783078368081475752296642517, 9.168699060246921232748010831277

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