Properties

Label 2-7360-1.1-c1-0-142
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  − 2.14·3-s + 5-s + 1.14·7-s + 1.60·9-s + 5.89·11-s + 4.89·13-s − 2.14·15-s − 5.89·17-s − 2.34·19-s − 2.45·21-s − 23-s + 25-s + 3.00·27-s − 3.74·29-s − 5.68·31-s − 12.6·33-s + 1.14·35-s − 4·37-s − 10.4·39-s − 1.05·41-s + 11.4·43-s + 1.60·45-s − 7.74·47-s − 5.68·49-s + 12.6·51-s − 12.9·53-s + 5.89·55-s + ⋯
L(s)  = 1  − 1.23·3-s + 0.447·5-s + 0.432·7-s + 0.533·9-s + 1.77·11-s + 1.35·13-s − 0.553·15-s − 1.42·17-s − 0.538·19-s − 0.536·21-s − 0.208·23-s + 0.200·25-s + 0.577·27-s − 0.695·29-s − 1.02·31-s − 2.20·33-s + 0.193·35-s − 0.657·37-s − 1.68·39-s − 0.165·41-s + 1.75·43-s + 0.238·45-s − 1.12·47-s − 0.812·49-s + 1.76·51-s − 1.78·53-s + 0.794·55-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 \( 1 + 2.14T + 3T^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 - 5.89T + 11T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 + 5.68T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 1.05T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 7.74T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 0.797T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 + 2.94T + 71T^{2} \)
73 \( 1 - 6.32T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 0.912T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−7.33376090863011107868147408649, −6.41544173273651734393787299669, −6.33302603927934448283351094776, −5.65517782601438631137421441792, −4.67647096424725255638775150203, −4.19068374196461224466068511198, −3.30040422528821150970767504078, −1.83638364502329388862135036578, −1.33945905739843124040361345645, 0, 1.33945905739843124040361345645, 1.83638364502329388862135036578, 3.30040422528821150970767504078, 4.19068374196461224466068511198, 4.67647096424725255638775150203, 5.65517782601438631137421441792, 6.33302603927934448283351094776, 6.41544173273651734393787299669, 7.33376090863011107868147408649

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