Properties

Label 2-7360-1.1-c1-0-50
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.43·3-s + 5-s + 3.08·7-s − 0.950·9-s + 6.46·11-s − 3.95·13-s − 1.43·15-s − 3.43·17-s − 3.08·19-s − 4.41·21-s − 23-s + 25-s + 5.65·27-s − 0.863·29-s − 5.95·31-s − 9.26·33-s + 3.08·35-s + 7.03·37-s + 5.65·39-s + 5.60·41-s − 8·43-s − 0.950·45-s + 3.90·47-s + 2.53·49-s + 4.91·51-s + 6·53-s + 6.46·55-s + ⋯
L(s)  = 1  − 0.826·3-s + 0.447·5-s + 1.16·7-s − 0.316·9-s + 1.95·11-s − 1.09·13-s − 0.369·15-s − 0.832·17-s − 0.708·19-s − 0.964·21-s − 0.208·23-s + 0.200·25-s + 1.08·27-s − 0.160·29-s − 1.06·31-s − 1.61·33-s + 0.521·35-s + 1.15·37-s + 0.905·39-s + 0.875·41-s − 1.21·43-s − 0.141·45-s + 0.569·47-s + 0.361·49-s + 0.687·51-s + 0.824·53-s + 0.872·55-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: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.767830785\)
\(L(\frac12)\) \(\approx\) \(1.767830785\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.43T + 3T^{2} \)
7 \( 1 - 3.08T + 7T^{2} \)
11 \( 1 - 6.46T + 11T^{2} \)
13 \( 1 + 3.95T + 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
29 \( 1 + 0.863T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 2.56T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−7.933205955193571812985985879215, −6.93153461407351123472291171001, −6.55920212659188382082974673358, −5.78651923104245643403329372777, −5.11294799573139502894573729276, −4.46751694572465156646127432685, −3.80678464993306920981008236594, −2.42302023027486780905590764989, −1.77116050911599780063759440301, −0.71188258339958861384617998866, 0.71188258339958861384617998866, 1.77116050911599780063759440301, 2.42302023027486780905590764989, 3.80678464993306920981008236594, 4.46751694572465156646127432685, 5.11294799573139502894573729276, 5.78651923104245643403329372777, 6.55920212659188382082974673358, 6.93153461407351123472291171001, 7.933205955193571812985985879215

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