Properties

Label 2-2280-1.1-c1-0-1
Degree $2$
Conductor $2280$
Sign $1$
Analytic cond. $18.2058$
Root an. cond. $4.26683$
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  − 3-s − 5-s − 3.60·7-s + 9-s − 1.60·11-s + 0.640·13-s + 15-s + 5.28·17-s + 19-s + 3.60·21-s − 8.24·23-s + 25-s − 27-s − 4.88·29-s − 7.28·31-s + 1.60·33-s + 3.60·35-s − 9.92·37-s − 0.640·39-s + 4.57·41-s + 7.92·43-s − 45-s + 0.249·47-s + 6.03·49-s − 5.28·51-s + 1.75·53-s + 1.60·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.36·7-s + 0.333·9-s − 0.485·11-s + 0.177·13-s + 0.258·15-s + 1.28·17-s + 0.229·19-s + 0.787·21-s − 1.72·23-s + 0.200·25-s − 0.192·27-s − 0.908·29-s − 1.30·31-s + 0.280·33-s + 0.610·35-s − 1.63·37-s − 0.102·39-s + 0.715·41-s + 1.20·43-s − 0.149·45-s + 0.0364·47-s + 0.861·49-s − 0.739·51-s + 0.240·53-s + 0.217·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(18.2058\)
Root analytic conductor: \(4.26683\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7711318550\)
\(L(\frac12)\) \(\approx\) \(0.7711318550\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 - 0.640T + 13T^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
23 \( 1 + 8.24T + 23T^{2} \)
29 \( 1 + 4.88T + 29T^{2} \)
31 \( 1 + 7.28T + 31T^{2} \)
37 \( 1 + 9.92T + 37T^{2} \)
41 \( 1 - 4.57T + 41T^{2} \)
43 \( 1 - 7.92T + 43T^{2} \)
47 \( 1 - 0.249T + 47T^{2} \)
53 \( 1 - 1.75T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 3.93T + 73T^{2} \)
79 \( 1 + 3.03T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 9.92T + 97T^{2} \)
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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

−9.159819905727666575048867224968, −8.116027610111035396979074700631, −7.40529565216148883397780339766, −6.70184761449653474416459435094, −5.75002105424631127636689219311, −5.35292408680954214046335593835, −3.86595299171695745071788446635, −3.52798538739300512165184170061, −2.18558175551321423842422298761, −0.56228806486245403795995779142, 0.56228806486245403795995779142, 2.18558175551321423842422298761, 3.52798538739300512165184170061, 3.86595299171695745071788446635, 5.35292408680954214046335593835, 5.75002105424631127636689219311, 6.70184761449653474416459435094, 7.40529565216148883397780339766, 8.116027610111035396979074700631, 9.159819905727666575048867224968

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