Properties

Label 2-9280-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s − 0.828·7-s + 9-s − 4.82·11-s + 2·13-s + 2·15-s − 2.82·17-s + 0.828·19-s + 1.65·21-s + 8.82·23-s + 25-s + 4·27-s − 29-s + 10.4·31-s + 9.65·33-s + 0.828·35-s − 8.48·37-s − 4·39-s − 6·41-s − 6·43-s − 45-s + 0.343·47-s − 6.31·49-s + 5.65·51-s − 7.65·53-s + 4.82·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 0.313·7-s + 0.333·9-s − 1.45·11-s + 0.554·13-s + 0.516·15-s − 0.685·17-s + 0.190·19-s + 0.361·21-s + 1.84·23-s + 0.200·25-s + 0.769·27-s − 0.185·29-s + 1.88·31-s + 1.68·33-s + 0.140·35-s − 1.39·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.149·45-s + 0.0500·47-s − 0.901·49-s + 0.792·51-s − 1.05·53-s + 0.651·55-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: \(0\)
Selberg data: \((2,\ 9280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4855184927\)
\(L(\frac12)\) \(\approx\) \(0.4855184927\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 3T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 0.828T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 + 7.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 7.65T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + 7.31T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 - 4.48T + 97T^{2} \)
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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

−7.59932487812108149441765818527, −6.87783239425211696312712032527, −6.37127013998031901860124061861, −5.63453358879925384067071873644, −4.85777586287927416587578480845, −4.65462151378253281909667432828, −3.24646832875909124368767226650, −2.88042051635152943426666082430, −1.48863754557940030231728729723, −0.36353875978956403783882246030, 0.36353875978956403783882246030, 1.48863754557940030231728729723, 2.88042051635152943426666082430, 3.24646832875909124368767226650, 4.65462151378253281909667432828, 4.85777586287927416587578480845, 5.63453358879925384067071873644, 6.37127013998031901860124061861, 6.87783239425211696312712032527, 7.59932487812108149441765818527

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