Properties

Label 2-6240-1.1-c1-0-86
Degree $2$
Conductor $6240$
Sign $-1$
Analytic cond. $49.8266$
Root an. cond. $7.05879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 1.56·7-s + 9-s − 2.43·11-s + 13-s + 15-s − 6.68·17-s + 3.12·19-s − 1.56·21-s + 4.68·23-s + 25-s + 27-s + 1.12·29-s − 8·31-s − 2.43·33-s − 1.56·35-s − 7.56·37-s + 39-s + 7.56·41-s − 0.876·43-s + 45-s − 10.2·47-s − 4.56·49-s − 6.68·51-s + 9.80·53-s − 2.43·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.590·7-s + 0.333·9-s − 0.735·11-s + 0.277·13-s + 0.258·15-s − 1.62·17-s + 0.716·19-s − 0.340·21-s + 0.976·23-s + 0.200·25-s + 0.192·27-s + 0.208·29-s − 1.43·31-s − 0.424·33-s − 0.263·35-s − 1.24·37-s + 0.160·39-s + 1.18·41-s − 0.133·43-s + 0.149·45-s − 1.49·47-s − 0.651·49-s − 0.936·51-s + 1.34·53-s − 0.328·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6240\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(49.8266\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 2.43T + 11T^{2} \)
17 \( 1 + 6.68T + 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 - 4.68T + 23T^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 7.56T + 37T^{2} \)
41 \( 1 - 7.56T + 41T^{2} \)
43 \( 1 + 0.876T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 9.80T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 3.56T + 61T^{2} \)
67 \( 1 - 2.24T + 67T^{2} \)
71 \( 1 + 0.684T + 71T^{2} \)
73 \( 1 - 8.24T + 73T^{2} \)
79 \( 1 + 8.68T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−7.66739975524794381391557172549, −6.96187486947307221570277406179, −6.44402167120497639031635131359, −5.48176086382127553624907816434, −4.87555161564514452426622178290, −3.90406283592132176560090088138, −3.09707261083776424017810187090, −2.42748496296284088419820241619, −1.48229324480088136082353184120, 0, 1.48229324480088136082353184120, 2.42748496296284088419820241619, 3.09707261083776424017810187090, 3.90406283592132176560090088138, 4.87555161564514452426622178290, 5.48176086382127553624907816434, 6.44402167120497639031635131359, 6.96187486947307221570277406179, 7.66739975524794381391557172549

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