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Properties

Label 2-2100-105.104-c1-0-32
Degree $2$
Conductor $2100$
Sign $0.962 + 0.270i$
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  + 1.73·3-s + (2.59 − 0.5i)7-s + 2.99·9-s + 1.73·13-s − 5.19i·19-s + (4.5 − 0.866i)21-s + 5.19·27-s + 1.73i·31-s − 10i·37-s + 2.99·39-s + 13i·43-s + (6.5 − 2.59i)49-s − 9i·57-s + 15.5i·61-s + (7.79 − 1.49i)63-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.981 − 0.188i)7-s + 0.999·9-s + 0.480·13-s − 1.19i·19-s + (0.981 − 0.188i)21-s + 1.00·27-s + 0.311i·31-s − 1.64i·37-s + 0.480·39-s + 1.98i·43-s + (0.928 − 0.371i)49-s − 1.19i·57-s + 1.99i·61-s + (0.981 − 0.188i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.051911383\)
\(L(\frac12)\) \(\approx\) \(3.051911383\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (-2.59 + 0.5i)T \)
good11 \( 1 - 11T^{2} \)
13 \( 1 - 1.73T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 5.19T + 97T^{2} \)
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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

−8.943201744343981376907722655309, −8.374849055208192769400894941796, −7.57733349804564331985869814190, −7.02477651547024199648754046865, −5.91626355514101721239787840292, −4.79990916377504536390143849766, −4.20525575450400196849538575753, −3.14861687764710790771742295915, −2.20074443196052952253487661255, −1.14080726101816712930625384144, 1.35482904634334075502487177937, 2.18285646763031811314215232010, 3.33190229275347937785287510868, 4.12752722639482743766759219506, 5.01410587650028801280534213848, 5.97629726608384868681022560228, 6.99608891321298122008330020627, 7.80956709302310985450596735412, 8.375909986747239195415640324602, 8.901863243884591967809903855557

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