Properties

Label 2-7360-1.1-c1-0-172
Degree $2$
Conductor $7360$
Sign $-1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s + 3·7-s + 9-s − 6·11-s − 4·13-s + 2·15-s + 7·17-s − 8·19-s + 6·21-s + 23-s + 25-s − 4·27-s − 3·29-s − 11·31-s − 12·33-s + 3·35-s − 7·37-s − 8·39-s + 5·41-s + 12·43-s + 45-s − 12·47-s + 2·49-s + 14·51-s − 7·53-s − 6·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.516·15-s + 1.69·17-s − 1.83·19-s + 1.30·21-s + 0.208·23-s + 1/5·25-s − 0.769·27-s − 0.557·29-s − 1.97·31-s − 2.08·33-s + 0.507·35-s − 1.15·37-s − 1.28·39-s + 0.780·41-s + 1.82·43-s + 0.149·45-s − 1.75·47-s + 2/7·49-s + 1.96·51-s − 0.961·53-s − 0.809·55-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: yes
Arithmetic: yes
Character: Trivial
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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.80444258438131855440369656099, −7.28032808378453838952195433447, −6.01180764604969203004967578598, −5.26274904697217525068619402733, −4.89403398734369608072549381313, −3.80471239000787530153835355849, −2.95252085158239399004331566056, −2.24124483468762206896946503341, −1.74629015271371498511355865696, 0, 1.74629015271371498511355865696, 2.24124483468762206896946503341, 2.95252085158239399004331566056, 3.80471239000787530153835355849, 4.89403398734369608072549381313, 5.26274904697217525068619402733, 6.01180764604969203004967578598, 7.28032808378453838952195433447, 7.80444258438131855440369656099

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