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Properties

Label 2-2100-35.3-c1-0-19
Degree $2$
Conductor $2100$
Sign $-0.657 + 0.753i$
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  + (−0.258 − 0.965i)3-s + (−1.15 + 2.38i)7-s + (−0.866 + 0.499i)9-s + (0.366 − 0.633i)11-s + (0.707 − 0.707i)13-s + (−3.15 + 0.845i)17-s + (−0.767 − 1.33i)19-s + (2.59 + 0.500i)21-s + (0.138 − 0.517i)23-s + (0.707 + 0.707i)27-s − 6.92i·29-s + (1.73 + i)31-s + (−0.707 − 0.189i)33-s + (−0.776 − 0.208i)37-s + (−0.866 − 0.500i)39-s + ⋯
L(s)  = 1  + (−0.149 − 0.557i)3-s + (−0.436 + 0.899i)7-s + (−0.288 + 0.166i)9-s + (0.110 − 0.191i)11-s + (0.196 − 0.196i)13-s + (−0.765 + 0.205i)17-s + (−0.176 − 0.305i)19-s + (0.566 + 0.109i)21-s + (0.0289 − 0.107i)23-s + (0.136 + 0.136i)27-s − 1.28i·29-s + (0.311 + 0.179i)31-s + (−0.123 − 0.0329i)33-s + (−0.127 − 0.0342i)37-s + (−0.138 − 0.0800i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.657 + 0.753i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7714215708\)
\(L(\frac12)\) \(\approx\) \(0.7714215708\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.15 - 2.38i)T \)
good11 \( 1 + (-0.366 + 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - 13iT^{2} \)
17 \( 1 + (3.15 - 0.845i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.767 + 1.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.138 + 0.517i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + (-1.73 - i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.776 + 0.208i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.19iT - 41T^{2} \)
43 \( 1 + (5.27 + 5.27i)T + 43iT^{2} \)
47 \( 1 + (0.0507 - 0.189i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.08 - 1.36i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.36 + 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.42 + 3.13i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.81 - 6.76i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (1.39 + 5.20i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.96 + 2.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.896 + 0.896i)T - 83iT^{2} \)
89 \( 1 + (5.36 + 9.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.39 + 7.39i)T + 97iT^{2} \)
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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

−8.675367202680760945040724936210, −8.221036293330300601763152930253, −7.13213098644319856403317639657, −6.43170073614995446159729355008, −5.80532945003641908298446731468, −4.93329041471648854860569716613, −3.79323042015675158660933695611, −2.72230520969319392359298530371, −1.88596477509227833786697171264, −0.28269942189794972665676783894, 1.30606674164815326137401842046, 2.79031220698645657984660821903, 3.75663927298155704793146193764, 4.44654177879821099317960366307, 5.28655050085486199043150193395, 6.40784785005321560502914215010, 6.88358698943048332925692581575, 7.85943067478011206269118416855, 8.716351358477484753206675676899, 9.493217813150092731680529794739

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