Properties

Label 2-7440-1.1-c1-0-105
Degree $2$
Conductor $7440$
Sign $-1$
Analytic cond. $59.4086$
Root an. cond. $7.70770$
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-s − 5-s + 3·7-s + 9-s − 3·11-s − 2·13-s − 15-s − 4·17-s + 3·19-s + 3·21-s − 5·23-s + 25-s + 27-s + 4·29-s − 31-s − 3·33-s − 3·35-s − 2·39-s + 4·41-s − 43-s − 45-s − 10·47-s + 2·49-s − 4·51-s + 3·53-s + 3·55-s + 3·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 0.258·15-s − 0.970·17-s + 0.688·19-s + 0.654·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.179·31-s − 0.522·33-s − 0.507·35-s − 0.320·39-s + 0.624·41-s − 0.152·43-s − 0.149·45-s − 1.45·47-s + 2/7·49-s − 0.560·51-s + 0.412·53-s + 0.404·55-s + 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-1$
Analytic conductor: \(59.4086\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7440} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7440,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
31 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−7.68725479977432822307577999288, −7.11807165283269808258545625927, −6.18728415879679334019480749094, −5.21113410915223105486668012874, −4.69052410990920949089855538752, −4.04295384300815496186729046136, −3.00861293584766238209711781996, −2.31300312121256825872872452868, −1.44006656415113705728781458911, 0, 1.44006656415113705728781458911, 2.31300312121256825872872452868, 3.00861293584766238209711781996, 4.04295384300815496186729046136, 4.69052410990920949089855538752, 5.21113410915223105486668012874, 6.18728415879679334019480749094, 7.11807165283269808258545625927, 7.68725479977432822307577999288

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