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Properties

Label 2-2100-35.27-c1-0-16
Degree $2$
Conductor $2100$
Sign $-0.861 + 0.507i$
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  + (−0.707 − 0.707i)3-s + (−1.71 + 2.01i)7-s + 1.00i·9-s − 4.27·11-s + (2.31 + 2.31i)13-s + (1.41 − 1.41i)17-s + 6.50·19-s + (2.63 − 0.209i)21-s + (−3.37 + 3.37i)23-s + (0.707 − 0.707i)27-s − 8.27i·29-s + 3.04i·31-s + (3.02 + 3.02i)33-s + (−7.68 − 7.68i)37-s − 3.27i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.648 + 0.760i)7-s + 0.333i·9-s − 1.28·11-s + (0.642 + 0.642i)13-s + (0.342 − 0.342i)17-s + 1.49·19-s + (0.575 − 0.0456i)21-s + (−0.704 + 0.704i)23-s + (0.136 − 0.136i)27-s − 1.53i·29-s + 0.546i·31-s + (0.526 + 0.526i)33-s + (−1.26 − 1.26i)37-s − 0.524i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.861 + 0.507i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3005160347\)
\(L(\frac12)\) \(\approx\) \(0.3005160347\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (1.71 - 2.01i)T \)
good11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + (-2.31 - 2.31i)T + 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 - 6.50T + 19T^{2} \)
23 \( 1 + (3.37 - 3.37i)T - 23iT^{2} \)
29 \( 1 + 8.27iT - 29T^{2} \)
31 \( 1 - 3.04iT - 31T^{2} \)
37 \( 1 + (7.68 + 7.68i)T + 37iT^{2} \)
41 \( 1 + 12.1iT - 41T^{2} \)
43 \( 1 + (5.53 - 5.53i)T - 43iT^{2} \)
47 \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \)
53 \( 1 + (8.61 - 8.61i)T - 53iT^{2} \)
59 \( 1 + 8.71T + 59T^{2} \)
61 \( 1 + 3.04iT - 61T^{2} \)
67 \( 1 + (-3.08 - 3.08i)T + 67iT^{2} \)
71 \( 1 - 1.72T + 71T^{2} \)
73 \( 1 + (7.07 + 7.07i)T + 73iT^{2} \)
79 \( 1 + 10.8iT - 79T^{2} \)
83 \( 1 + (1.02 + 1.02i)T + 83iT^{2} \)
89 \( 1 + 7.88T + 89T^{2} \)
97 \( 1 + (-1.92 + 1.92i)T - 97iT^{2} \)
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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

−8.844801457443832920741726585212, −7.81916313978164875875816006038, −7.34042874178777433046598687764, −6.27873118746722293131166049201, −5.64640766954909005967424484321, −5.04173060919025202023656733916, −3.69173865939477048773416045634, −2.78740687366187929745922197608, −1.73707251094916400993795965929, −0.11721476503325082718321279181, 1.25565145232932968965167172631, 3.03157186704621543349808748703, 3.49293958100275916774744799470, 4.72770455178827514802900274293, 5.39129003859181140759402028335, 6.22912097350820945807366307665, 7.06540493484782491566415352852, 7.909119005061102636619518156330, 8.547701802641047782246683582426, 9.817884804652893594707778755097

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