Properties

Label 2-3040-1.1-c1-0-71
Degree $2$
Conductor $3040$
Sign $-1$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
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  + 2.21·3-s + 5-s − 1.52·7-s + 1.90·9-s − 2.90·11-s − 1.31·13-s + 2.21·15-s − 7.80·17-s − 19-s − 3.37·21-s − 4.28·23-s + 25-s − 2.42·27-s − 9.18·29-s + 7.80·31-s − 6.42·33-s − 1.52·35-s + 8.16·37-s − 2.90·39-s − 11.0·41-s − 1.09·43-s + 1.90·45-s − 4.28·47-s − 4.67·49-s − 17.2·51-s + 5.54·53-s − 2.90·55-s + ⋯
L(s)  = 1  + 1.27·3-s + 0.447·5-s − 0.576·7-s + 0.634·9-s − 0.875·11-s − 0.363·13-s + 0.571·15-s − 1.89·17-s − 0.229·19-s − 0.737·21-s − 0.892·23-s + 0.200·25-s − 0.467·27-s − 1.70·29-s + 1.40·31-s − 1.11·33-s − 0.257·35-s + 1.34·37-s − 0.464·39-s − 1.72·41-s − 0.167·43-s + 0.283·45-s − 0.624·47-s − 0.667·49-s − 2.42·51-s + 0.761·53-s − 0.391·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 + T \)
good3 \( 1 - 2.21T + 3T^{2} \)
7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 + 1.31T + 13T^{2} \)
17 \( 1 + 7.80T + 17T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + 9.18T + 29T^{2} \)
31 \( 1 - 7.80T + 31T^{2} \)
37 \( 1 - 8.16T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 1.09T + 43T^{2} \)
47 \( 1 + 4.28T + 47T^{2} \)
53 \( 1 - 5.54T + 53T^{2} \)
59 \( 1 - 1.86T + 59T^{2} \)
61 \( 1 - 0.709T + 61T^{2} \)
67 \( 1 + 1.45T + 67T^{2} \)
71 \( 1 + 2.29T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 6.13T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 3.67T + 89T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−8.302244886292726521476238293675, −7.85895850653421738583582495265, −6.84900100655481773393968362940, −6.23274062207919122270060707047, −5.19124563299928196332641251004, −4.27532332749531487902873461989, −3.38440485260519177232508606994, −2.46326495125272572352308101429, −2.02557223296903036683766298670, 0, 2.02557223296903036683766298670, 2.46326495125272572352308101429, 3.38440485260519177232508606994, 4.27532332749531487902873461989, 5.19124563299928196332641251004, 6.23274062207919122270060707047, 6.84900100655481773393968362940, 7.85895850653421738583582495265, 8.302244886292726521476238293675

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