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Properties

Label 2-2100-21.20-c1-0-46
Degree $2$
Conductor $2100$
Sign $-0.281 + 0.959i$
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  + (1.30 − 1.14i)3-s + (2.23 − 1.41i)7-s + (0.381 − 2.97i)9-s − 1.13i·11-s + 0.333i·13-s − 4.20·17-s − 1.95i·19-s + (1.28 − 4.39i)21-s − 2.54i·23-s + (−2.90 − 4.30i)27-s − 8.49i·29-s + 5.11i·31-s + (−1.30 − 1.47i)33-s + 2.23·37-s + (0.381 + 0.434i)39-s + ⋯
L(s)  = 1  + (0.750 − 0.660i)3-s + (0.845 − 0.534i)7-s + (0.127 − 0.991i)9-s − 0.342i·11-s + 0.0925i·13-s − 1.02·17-s − 0.448i·19-s + (0.281 − 0.959i)21-s − 0.529i·23-s + (−0.559 − 0.828i)27-s − 1.57i·29-s + 0.918i·31-s + (−0.226 − 0.257i)33-s + 0.367·37-s + (0.0611 + 0.0695i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.281 + 0.959i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.286630324\)
\(L(\frac12)\) \(\approx\) \(2.286630324\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 1.14i)T \)
5 \( 1 \)
7 \( 1 + (-2.23 + 1.41i)T \)
good11 \( 1 + 1.13iT - 11T^{2} \)
13 \( 1 - 0.333iT - 13T^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 + 1.95iT - 19T^{2} \)
23 \( 1 + 2.54iT - 23T^{2} \)
29 \( 1 + 8.49iT - 29T^{2} \)
31 \( 1 - 5.11iT - 31T^{2} \)
37 \( 1 - 2.23T + 37T^{2} \)
41 \( 1 - 5.81T + 41T^{2} \)
43 \( 1 - 0.527T + 43T^{2} \)
47 \( 1 + 7.42T + 47T^{2} \)
53 \( 1 + 8.22iT - 53T^{2} \)
59 \( 1 - 3.59T + 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 - 6.70T + 67T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 - 1.41iT - 73T^{2} \)
79 \( 1 - 1.47T + 79T^{2} \)
83 \( 1 + 5.81T + 83T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + 9.69iT - 97T^{2} \)
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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

−8.545550792171915609784282926004, −8.292668620317885082037017796855, −7.31069519413385106653436535704, −6.76260657338250851295155303556, −5.84517523908396294789923147617, −4.63146623448464342206982687275, −3.97105753935501377602673779320, −2.77984337197918813973976682376, −1.92387011213788899851647357367, −0.72218219357845363126121650119, 1.67758239293601499710818808470, 2.51090358033144846788020755574, 3.57712047952392315022503886687, 4.51962072900830305611488616641, 5.11001642888165196150225967491, 6.07401370131897525651161354740, 7.24309644826617421174197234080, 7.935186160235082200799233923619, 8.621361897476690871177384857876, 9.267252499774271321644280675599

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