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Properties

Label 2-2100-15.2-c1-0-19
Degree $2$
Conductor $2100$
Sign $0.892 + 0.451i$
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  + (0.573 − 1.63i)3-s + (0.707 + 0.707i)7-s + (−2.34 − 1.87i)9-s + 6.10i·11-s + (2.13 − 2.13i)13-s + (1.21 − 1.21i)17-s − 0.390i·19-s + (1.56 − 0.750i)21-s + (2.04 + 2.04i)23-s + (−4.40 + 2.75i)27-s + 5.67·29-s + 1.37·31-s + (9.97 + 3.50i)33-s + (2.84 + 2.84i)37-s + (−2.26 − 4.71i)39-s + ⋯
L(s)  = 1  + (0.331 − 0.943i)3-s + (0.267 + 0.267i)7-s + (−0.780 − 0.624i)9-s + 1.84i·11-s + (0.591 − 0.591i)13-s + (0.294 − 0.294i)17-s − 0.0895i·19-s + (0.340 − 0.163i)21-s + (0.426 + 0.426i)23-s + (−0.848 + 0.529i)27-s + 1.05·29-s + 0.246·31-s + (1.73 + 0.609i)33-s + (0.467 + 0.467i)37-s + (−0.362 − 0.754i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.892 + 0.451i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.096615346\)
\(L(\frac12)\) \(\approx\) \(2.096615346\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.573 + 1.63i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 6.10iT - 11T^{2} \)
13 \( 1 + (-2.13 + 2.13i)T - 13iT^{2} \)
17 \( 1 + (-1.21 + 1.21i)T - 17iT^{2} \)
19 \( 1 + 0.390iT - 19T^{2} \)
23 \( 1 + (-2.04 - 2.04i)T + 23iT^{2} \)
29 \( 1 - 5.67T + 29T^{2} \)
31 \( 1 - 1.37T + 31T^{2} \)
37 \( 1 + (-2.84 - 2.84i)T + 37iT^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + (-3.04 + 3.04i)T - 43iT^{2} \)
47 \( 1 + (-1.32 + 1.32i)T - 47iT^{2} \)
53 \( 1 + (-9.10 - 9.10i)T + 53iT^{2} \)
59 \( 1 - 6.21T + 59T^{2} \)
61 \( 1 + 6.30T + 61T^{2} \)
67 \( 1 + (-7.63 - 7.63i)T + 67iT^{2} \)
71 \( 1 - 6.26iT - 71T^{2} \)
73 \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \)
79 \( 1 + 6.56iT - 79T^{2} \)
83 \( 1 + (3.59 + 3.59i)T + 83iT^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \)
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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

−8.910871129089710459943078977821, −8.211558121447056755066772285568, −7.39028268441892576047573212713, −6.95912154000483020750270222915, −5.95716861565561871050113358088, −5.13180626397820321178126519467, −4.12425794157643661170976305465, −2.92588753114784464806959016014, −2.09957802379395000116897410238, −1.03198486421328064012311248750, 0.931718425877602922748575484738, 2.56000046363310483899414289051, 3.46834598050222002010016489495, 4.13424943855485990945733894642, 5.10915312292137707880418692784, 5.92315163977392665403000573436, 6.66648250367490379265809001937, 8.030404374516675633810432381886, 8.397055339551993042838294116913, 9.079495098991529830042823194962

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