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Properties

Label 2-2100-105.104-c1-0-43
Degree $2$
Conductor $2100$
Sign $-0.383 + 0.923i$
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 + (−1.73 − 2i)7-s + 2.99·9-s − 6.92·13-s − 3.46i·19-s + (−2.99 − 3.46i)21-s + 5.19·27-s − 10.3i·31-s − 10i·37-s − 11.9·39-s − 8i·43-s + (−1.00 + 6.92i)49-s − 5.99i·57-s − 6.92i·61-s + (−5.19 − 5.99i)63-s + ⋯
L(s)  = 1  + 1.00·3-s + (−0.654 − 0.755i)7-s + 0.999·9-s − 1.92·13-s − 0.794i·19-s + (−0.654 − 0.755i)21-s + 1.00·27-s − 1.86i·31-s − 1.64i·37-s − 1.92·39-s − 1.21i·43-s + (−0.142 + 0.989i)49-s − 0.794i·57-s − 0.887i·61-s + (−0.654 − 0.755i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.383 + 0.923i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.383 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.522111532\)
\(L(\frac12)\) \(\approx\) \(1.522111532\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2i)T \)
good11 \( 1 - 11T^{2} \)
13 \( 1 + 6.92T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−9.029768796894498315420059931662, −7.966386323861606859444634637562, −7.25920638089283222529635338390, −6.94835034743006451447533808283, −5.65539853887124350186604391896, −4.55122546512106382630636366765, −3.92624516817552349777897170739, −2.83706573276521949486431752459, −2.14869935572630179401484384735, −0.43498140567954278433299108874, 1.65979988973819904990125479196, 2.73157253463248105314251629433, 3.23788762924008560881222829067, 4.50230874963391108480892462594, 5.21557276993640092468343519991, 6.38361269883548258548127153238, 7.09759146435658612902322501567, 7.88350057493498853208473744754, 8.597808230211660468300439526366, 9.379860303618144096162492283288

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