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Properties

Label 2-1680-420.419-c0-0-8
Degree $2$
Conductor $1680$
Sign $1$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $0$
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 − 5-s + 7-s + 9-s − 15-s + 21-s + 25-s + 27-s − 35-s + 2·41-s − 2·43-s − 45-s − 2·47-s + 49-s + 63-s + 2·67-s + 75-s + 81-s − 2·83-s − 2·89-s + 2·101-s − 105-s − 2·109-s + ⋯
L(s)  = 1  + 3-s − 5-s + 7-s + 9-s − 15-s + 21-s + 25-s + 27-s − 35-s + 2·41-s − 2·43-s − 45-s − 2·47-s + 49-s + 63-s + 2·67-s + 75-s + 81-s − 2·83-s − 2·89-s + 2·101-s − 105-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1680} (1679, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520676479\)
\(L(\frac12)\) \(\approx\) \(1.520676479\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 + T )^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T^{2} \)
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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

−9.406881174600087443002462705729, −8.462567800048546890142510140825, −8.128483073749585974951947044333, −7.40536831019374998333568504760, −6.62818564967286491094144000920, −5.18636394466576954452610109083, −4.41278106690900113479546324313, −3.65306768974579641236280726435, −2.64614589374001882568057576095, −1.42742505967143579043811336925, 1.42742505967143579043811336925, 2.64614589374001882568057576095, 3.65306768974579641236280726435, 4.41278106690900113479546324313, 5.18636394466576954452610109083, 6.62818564967286491094144000920, 7.40536831019374998333568504760, 8.128483073749585974951947044333, 8.462567800048546890142510140825, 9.406881174600087443002462705729

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