Properties

Label 2-7360-1.1-c1-0-113
Degree $2$
Conductor $7360$
Sign $-1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
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  − 1.93·3-s − 5-s + 2.47·7-s + 0.729·9-s + 4.45·11-s − 4.02·13-s + 1.93·15-s + 1.38·17-s − 4.02·19-s − 4.77·21-s − 23-s + 25-s + 4.38·27-s + 2.23·29-s + 5.15·31-s − 8.60·33-s − 2.47·35-s − 7.06·37-s + 7.76·39-s + 5.19·41-s − 9.65·43-s − 0.729·45-s − 7.27·47-s − 0.884·49-s − 2.67·51-s + 0.229·53-s − 4.45·55-s + ⋯
L(s)  = 1  − 1.11·3-s − 0.447·5-s + 0.934·7-s + 0.243·9-s + 1.34·11-s − 1.11·13-s + 0.498·15-s + 0.336·17-s − 0.923·19-s − 1.04·21-s − 0.208·23-s + 0.200·25-s + 0.843·27-s + 0.414·29-s + 0.925·31-s − 1.49·33-s − 0.418·35-s − 1.16·37-s + 1.24·39-s + 0.811·41-s − 1.47·43-s − 0.108·45-s − 1.06·47-s − 0.126·49-s − 0.374·51-s + 0.0315·53-s − 0.600·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7360\)    =    \(2^{6} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(58.7698\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 + 1.93T + 3T^{2} \)
7 \( 1 - 2.47T + 7T^{2} \)
11 \( 1 - 4.45T + 11T^{2} \)
13 \( 1 + 4.02T + 13T^{2} \)
17 \( 1 - 1.38T + 17T^{2} \)
19 \( 1 + 4.02T + 19T^{2} \)
29 \( 1 - 2.23T + 29T^{2} \)
31 \( 1 - 5.15T + 31T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 + 7.27T + 47T^{2} \)
53 \( 1 - 0.229T + 53T^{2} \)
59 \( 1 + 6.09T + 59T^{2} \)
61 \( 1 - 8.22T + 61T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 + 0.318T + 71T^{2} \)
73 \( 1 - 5.47T + 73T^{2} \)
79 \( 1 - 1.75T + 79T^{2} \)
83 \( 1 - 5.67T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−7.51532623455609868306879028719, −6.59495905303927343856154244085, −6.37822069463982747483907772624, −5.28179934534914695750520335478, −4.81027566171375787456784108570, −4.22034477990801544761770100544, −3.23914779977114554770265087481, −2.05926238088964601160030758814, −1.12573979619492105489395744133, 0, 1.12573979619492105489395744133, 2.05926238088964601160030758814, 3.23914779977114554770265087481, 4.22034477990801544761770100544, 4.81027566171375787456784108570, 5.28179934534914695750520335478, 6.37822069463982747483907772624, 6.59495905303927343856154244085, 7.51532623455609868306879028719

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