Properties

Label 2-1680-1.1-c3-0-39
Degree $2$
Conductor $1680$
Sign $-1$
Analytic cond. $99.1232$
Root an. cond. $9.95606$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5·5-s − 7·7-s + 9·9-s − 24.5·11-s − 35.0·13-s + 15·15-s − 18.4·17-s + 67.4·19-s + 21·21-s + 145.·23-s + 25·25-s − 27·27-s + 214.·29-s + 88.6·31-s + 73.7·33-s + 35·35-s + 162.·37-s + 105.·39-s − 337.·41-s − 122.·43-s − 45·45-s − 354.·47-s + 49·49-s + 55.2·51-s + 676.·53-s + 122.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s − 0.674·11-s − 0.747·13-s + 0.258·15-s − 0.262·17-s + 0.813·19-s + 0.218·21-s + 1.32·23-s + 0.200·25-s − 0.192·27-s + 1.37·29-s + 0.513·31-s + 0.389·33-s + 0.169·35-s + 0.720·37-s + 0.431·39-s − 1.28·41-s − 0.433·43-s − 0.149·45-s − 1.09·47-s + 0.142·49-s + 0.151·51-s + 1.75·53-s + 0.301·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(99.1232\)
Root analytic conductor: \(9.95606\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1680,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
5 \( 1 + 5T \)
7 \( 1 + 7T \)
good11 \( 1 + 24.5T + 1.33e3T^{2} \)
13 \( 1 + 35.0T + 2.19e3T^{2} \)
17 \( 1 + 18.4T + 4.91e3T^{2} \)
19 \( 1 - 67.4T + 6.85e3T^{2} \)
23 \( 1 - 145.T + 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 - 88.6T + 2.97e4T^{2} \)
37 \( 1 - 162.T + 5.06e4T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 + 122.T + 7.95e4T^{2} \)
47 \( 1 + 354.T + 1.03e5T^{2} \)
53 \( 1 - 676.T + 1.48e5T^{2} \)
59 \( 1 + 501.T + 2.05e5T^{2} \)
61 \( 1 + 708.T + 2.26e5T^{2} \)
67 \( 1 - 907.T + 3.00e5T^{2} \)
71 \( 1 + 430.T + 3.57e5T^{2} \)
73 \( 1 - 41.3T + 3.89e5T^{2} \)
79 \( 1 + 890.T + 4.93e5T^{2} \)
83 \( 1 - 1.05e3T + 5.71e5T^{2} \)
89 \( 1 - 1.47e3T + 7.04e5T^{2} \)
97 \( 1 - 555.T + 9.12e5T^{2} \)
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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

−8.556201262653392522003485250867, −7.68966543785456998591891239595, −6.98447691886513795193893908172, −6.25501745590821073112018368398, −5.10018135428180571689261152901, −4.71814790612179496745598998928, −3.40296048652726666401617988475, −2.57651878391400688897358158339, −1.05093785090057980065476714016, 0, 1.05093785090057980065476714016, 2.57651878391400688897358158339, 3.40296048652726666401617988475, 4.71814790612179496745598998928, 5.10018135428180571689261152901, 6.25501745590821073112018368398, 6.98447691886513795193893908172, 7.68966543785456998591891239595, 8.556201262653392522003485250867

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