Properties

Label 2-7360-1.1-c1-0-174
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  + 2.79·3-s + 5-s − 1.79·7-s + 4.79·9-s + 0.791·11-s − 5.79·13-s + 2.79·15-s + 0.791·17-s − 5.79·19-s − 5·21-s + 23-s + 25-s + 4.99·27-s − 7.58·29-s − 3.37·31-s + 2.20·33-s − 1.79·35-s + 4·37-s − 16.1·39-s − 6.79·41-s − 11.1·43-s + 4.79·45-s − 4.41·47-s − 3.79·49-s + 2.20·51-s − 6·53-s + 0.791·55-s + ⋯
L(s)  = 1  + 1.61·3-s + 0.447·5-s − 0.677·7-s + 1.59·9-s + 0.238·11-s − 1.60·13-s + 0.720·15-s + 0.191·17-s − 1.32·19-s − 1.09·21-s + 0.208·23-s + 0.200·25-s + 0.962·27-s − 1.40·29-s − 0.605·31-s + 0.384·33-s − 0.302·35-s + 0.657·37-s − 2.58·39-s − 1.06·41-s − 1.70·43-s + 0.714·45-s − 0.644·47-s − 0.541·49-s + 0.309·51-s − 0.824·53-s + 0.106·55-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 \( 1 - 2.79T + 3T^{2} \)
7 \( 1 + 1.79T + 7T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 + 5.79T + 13T^{2} \)
17 \( 1 - 0.791T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 7.95T + 97T^{2} \)
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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

−7.72559270071860145983813959721, −6.90538024139718686630943959359, −6.45618258307858548893260869478, −5.33093501242736514766254819710, −4.59617821246778085282063405507, −3.66439985493917045119517114312, −3.13068602656314140045396435743, −2.24434515295162159333032384585, −1.79253750151979273035453353504, 0, 1.79253750151979273035453353504, 2.24434515295162159333032384585, 3.13068602656314140045396435743, 3.66439985493917045119517114312, 4.59617821246778085282063405507, 5.33093501242736514766254819710, 6.45618258307858548893260869478, 6.90538024139718686630943959359, 7.72559270071860145983813959721

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