Properties

Label 2-9280-1.1-c1-0-113
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  + 1.70·3-s − 5-s + 3.70·7-s − 0.0783·9-s + 0.630·11-s + 4.34·13-s − 1.70·15-s − 1.55·17-s + 5.70·19-s + 6.34·21-s + 6.63·23-s + 25-s − 5.26·27-s + 29-s − 2.29·31-s + 1.07·33-s − 3.70·35-s + 2.44·37-s + 7.41·39-s + 5.60·41-s − 12.5·43-s + 0.0783·45-s + 2.29·47-s + 6.75·49-s − 2.65·51-s − 0.921·53-s − 0.630·55-s + ⋯
L(s)  = 1  + 0.986·3-s − 0.447·5-s + 1.40·7-s − 0.0261·9-s + 0.190·11-s + 1.20·13-s − 0.441·15-s − 0.376·17-s + 1.30·19-s + 1.38·21-s + 1.38·23-s + 0.200·25-s − 1.01·27-s + 0.185·29-s − 0.411·31-s + 0.187·33-s − 0.626·35-s + 0.402·37-s + 1.18·39-s + 0.874·41-s − 1.91·43-s + 0.0116·45-s + 0.334·47-s + 0.965·49-s − 0.371·51-s − 0.126·53-s − 0.0850·55-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

\(L(1)\) \(\approx\) \(3.815110948\)
\(L(\frac12)\) \(\approx\) \(3.815110948\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.70T + 7T^{2} \)
11 \( 1 - 0.630T + 11T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 - 5.70T + 19T^{2} \)
23 \( 1 - 6.63T + 23T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 - 2.44T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 2.29T + 47T^{2} \)
53 \( 1 + 0.921T + 53T^{2} \)
59 \( 1 - 3.60T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 3.12T + 83T^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−7.78564079554128241044812528589, −7.32453090223226941010210005615, −6.45926556688891729312368073593, −5.47471492372915927988487682742, −4.94598658445795776316211612627, −4.05980441164129616465328405337, −3.42925222716854963684195562971, −2.71571850331521682898704755128, −1.71699189980453273525072295488, −0.968173706905548842808269452568, 0.968173706905548842808269452568, 1.71699189980453273525072295488, 2.71571850331521682898704755128, 3.42925222716854963684195562971, 4.05980441164129616465328405337, 4.94598658445795776316211612627, 5.47471492372915927988487682742, 6.45926556688891729312368073593, 7.32453090223226941010210005615, 7.78564079554128241044812528589

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