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Properties

Label 2-7840-1.1-c1-0-161
Degree $2$
Conductor $7840$
Sign $-1$
Analytic cond. $62.6027$
Root an. cond. $7.91218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s + 6·9-s − 3·11-s − 13-s − 3·15-s + 17-s − 4·19-s + 4·23-s + 25-s + 9·27-s − 9·29-s − 6·31-s − 9·33-s − 8·37-s − 3·39-s − 6·41-s − 8·43-s − 6·45-s + 7·47-s + 3·51-s − 8·53-s + 3·55-s − 12·57-s − 4·59-s − 10·61-s + 65-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 2·9-s − 0.904·11-s − 0.277·13-s − 0.774·15-s + 0.242·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s − 1.07·31-s − 1.56·33-s − 1.31·37-s − 0.480·39-s − 0.937·41-s − 1.21·43-s − 0.894·45-s + 1.02·47-s + 0.420·51-s − 1.09·53-s + 0.404·55-s − 1.58·57-s − 0.520·59-s − 1.28·61-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7840\)    =    \(2^{5} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(62.6027\)
Root analytic conductor: \(7.91218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−7.40357809070726097271948588680, −7.39623366266840898901472779294, −6.26579001325253428869794121445, −5.17250506877976527900763571362, −4.57346402623878792114980623657, −3.46514829938570680088967333396, −3.35672458975389467409403549909, −2.25825293857785414994283197016, −1.69163648269613371170065700417, 0, 1.69163648269613371170065700417, 2.25825293857785414994283197016, 3.35672458975389467409403549909, 3.46514829938570680088967333396, 4.57346402623878792114980623657, 5.17250506877976527900763571362, 6.26579001325253428869794121445, 7.39623366266840898901472779294, 7.40357809070726097271948588680

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