Properties

Label 2-7360-1.1-c1-0-114
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  + 0.706·3-s − 5-s − 5.21·7-s − 2.50·9-s + 4.73·11-s + 0.250·13-s − 0.706·15-s − 2.11·17-s + 5.84·19-s − 3.68·21-s − 23-s + 25-s − 3.88·27-s + 6.19·29-s + 3.60·31-s + 3.34·33-s + 5.21·35-s − 10.8·37-s + 0.177·39-s + 1.02·41-s + 11.6·43-s + 2.50·45-s − 0.134·47-s + 20.1·49-s − 1.49·51-s + 9.78·53-s − 4.73·55-s + ⋯
L(s)  = 1  + 0.408·3-s − 0.447·5-s − 1.97·7-s − 0.833·9-s + 1.42·11-s + 0.0696·13-s − 0.182·15-s − 0.512·17-s + 1.34·19-s − 0.803·21-s − 0.208·23-s + 0.200·25-s − 0.748·27-s + 1.14·29-s + 0.648·31-s + 0.582·33-s + 0.881·35-s − 1.78·37-s + 0.0284·39-s + 0.159·41-s + 1.77·43-s + 0.372·45-s − 0.0196·47-s + 2.88·49-s − 0.209·51-s + 1.34·53-s − 0.638·55-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 \( 1 - 0.706T + 3T^{2} \)
7 \( 1 + 5.21T + 7T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 - 0.250T + 13T^{2} \)
17 \( 1 + 2.11T + 17T^{2} \)
19 \( 1 - 5.84T + 19T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 - 3.60T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 1.02T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + 0.134T + 47T^{2} \)
53 \( 1 - 9.78T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 1.05T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 6.98T + 73T^{2} \)
79 \( 1 - 6.07T + 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 - 0.300T + 89T^{2} \)
97 \( 1 - 0.307T + 97T^{2} \)
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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

−7.34012337177708709766478963655, −6.94870543560697958220498147322, −6.14096002096924923883856894260, −5.75695864828326471346046780575, −4.48820923138682718976706978380, −3.73256099476610854908726656479, −3.17033281947353231456299148302, −2.60974333100114836473340189767, −1.12771983226195061090957751720, 0, 1.12771983226195061090957751720, 2.60974333100114836473340189767, 3.17033281947353231456299148302, 3.73256099476610854908726656479, 4.48820923138682718976706978380, 5.75695864828326471346046780575, 6.14096002096924923883856894260, 6.94870543560697958220498147322, 7.34012337177708709766478963655

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