Properties

Label 2-2100-1.1-c1-0-7
Degree $2$
Conductor $2100$
Sign $1$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 4·11-s + 6·13-s − 2·17-s + 6·19-s − 21-s − 2·23-s + 27-s + 6·29-s − 2·31-s − 4·33-s + 4·37-s + 6·39-s + 8·41-s + 4·43-s − 4·47-s + 49-s − 2·51-s − 6·53-s + 6·57-s + 4·59-s + 14·61-s − 63-s − 4·67-s − 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1.37·19-s − 0.218·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.657·37-s + 0.960·39-s + 1.24·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.794·57-s + 0.520·59-s + 1.79·61-s − 0.125·63-s − 0.488·67-s − 0.240·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2100} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.155871180\)
\(L(\frac12)\) \(\approx\) \(2.155871180\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.116671878067058228668736642197, −8.217274976762957722158366546561, −7.79475966786471025761067987992, −6.77238615811618901030894496151, −5.97716076552890173486392763311, −5.12771157921464522957188091436, −4.03694670278567160946220010364, −3.22455670944000123351362763963, −2.37205064712529596769921999280, −0.973469207532226085894380069823, 0.973469207532226085894380069823, 2.37205064712529596769921999280, 3.22455670944000123351362763963, 4.03694670278567160946220010364, 5.12771157921464522957188091436, 5.97716076552890173486392763311, 6.77238615811618901030894496151, 7.79475966786471025761067987992, 8.217274976762957722158366546561, 9.116671878067058228668736642197

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