Properties

Label 2-9280-1.1-c1-0-207
Degree $2$
Conductor $9280$
Sign $-1$
Analytic cond. $74.1011$
Root an. cond. $8.60820$
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  + 1.70·3-s + 5-s − 0.630·7-s − 0.0783·9-s + 0.290·11-s + 0.921·13-s + 1.70·15-s + 4.97·17-s − 6.04·19-s − 1.07·21-s − 2.29·23-s + 25-s − 5.26·27-s − 29-s − 10.0·31-s + 0.496·33-s − 0.630·35-s − 1.55·37-s + 1.57·39-s + 0.340·41-s − 5.70·43-s − 0.0783·45-s + 1.12·47-s − 6.60·49-s + 8.49·51-s + 0.340·53-s + 0.290·55-s + ⋯
L(s)  = 1  + 0.986·3-s + 0.447·5-s − 0.238·7-s − 0.0261·9-s + 0.0876·11-s + 0.255·13-s + 0.441·15-s + 1.20·17-s − 1.38·19-s − 0.235·21-s − 0.477·23-s + 0.200·25-s − 1.01·27-s − 0.185·29-s − 1.80·31-s + 0.0865·33-s − 0.106·35-s − 0.255·37-s + 0.252·39-s + 0.0531·41-s − 0.870·43-s − 0.0116·45-s + 0.164·47-s − 0.943·49-s + 1.18·51-s + 0.0467·53-s + 0.0392·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9280\)    =    \(2^{6} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(74.1011\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9280,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
29 \( 1 + T \)
good3 \( 1 - 1.70T + 3T^{2} \)
7 \( 1 + 0.630T + 7T^{2} \)
11 \( 1 - 0.290T + 11T^{2} \)
13 \( 1 - 0.921T + 13T^{2} \)
17 \( 1 - 4.97T + 17T^{2} \)
19 \( 1 + 6.04T + 19T^{2} \)
23 \( 1 + 2.29T + 23T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 - 0.340T + 41T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 - 1.12T + 47T^{2} \)
53 \( 1 - 0.340T + 53T^{2} \)
59 \( 1 - 9.75T + 59T^{2} \)
61 \( 1 + 3.07T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 + 9.07T + 71T^{2} \)
73 \( 1 + 6.94T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 - 4.73T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−7.50571174010449669533681410837, −6.75010693772042824110380954734, −5.91642843499371259951119453733, −5.49605422879285073542749374396, −4.42479689717374452642519478690, −3.62219220649379881599343472504, −3.11178995129076700574348379467, −2.17907675538213825529720609473, −1.54907339498591225861231479249, 0, 1.54907339498591225861231479249, 2.17907675538213825529720609473, 3.11178995129076700574348379467, 3.62219220649379881599343472504, 4.42479689717374452642519478690, 5.49605422879285073542749374396, 5.91642843499371259951119453733, 6.75010693772042824110380954734, 7.50571174010449669533681410837

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