Properties

Label 2-4015-1.1-c1-0-194
Degree $2$
Conductor $4015$
Sign $-1$
Analytic cond. $32.0599$
Root an. cond. $5.66214$
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  − 2.71·2-s + 1.57·3-s + 5.37·4-s + 5-s − 4.26·6-s + 1.27·7-s − 9.15·8-s − 0.531·9-s − 2.71·10-s + 11-s + 8.43·12-s + 1.99·13-s − 3.47·14-s + 1.57·15-s + 14.1·16-s − 4.79·17-s + 1.44·18-s − 0.355·19-s + 5.37·20-s + 2.00·21-s − 2.71·22-s + 1.37·23-s − 14.3·24-s + 25-s − 5.40·26-s − 5.54·27-s + 6.87·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 0.907·3-s + 2.68·4-s + 0.447·5-s − 1.74·6-s + 0.483·7-s − 3.23·8-s − 0.177·9-s − 0.858·10-s + 0.301·11-s + 2.43·12-s + 0.552·13-s − 0.928·14-s + 0.405·15-s + 3.52·16-s − 1.16·17-s + 0.340·18-s − 0.0816·19-s + 1.20·20-s + 0.438·21-s − 0.578·22-s + 0.287·23-s − 2.93·24-s + 0.200·25-s − 1.06·26-s − 1.06·27-s + 1.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $-1$
Analytic conductor: \(32.0599\)
Root analytic conductor: \(5.66214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4015,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 + T \)
good2 \( 1 + 2.71T + 2T^{2} \)
3 \( 1 - 1.57T + 3T^{2} \)
7 \( 1 - 1.27T + 7T^{2} \)
13 \( 1 - 1.99T + 13T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
19 \( 1 + 0.355T + 19T^{2} \)
23 \( 1 - 1.37T + 23T^{2} \)
29 \( 1 + 9.65T + 29T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 - 1.99T + 37T^{2} \)
41 \( 1 + 5.40T + 41T^{2} \)
43 \( 1 + 5.32T + 43T^{2} \)
47 \( 1 - 8.84T + 47T^{2} \)
53 \( 1 + 7.05T + 53T^{2} \)
59 \( 1 + 3.35T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 8.88T + 67T^{2} \)
71 \( 1 + 8.65T + 71T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 + 6.54T + 83T^{2} \)
89 \( 1 - 0.603T + 89T^{2} \)
97 \( 1 + 2.67T + 97T^{2} \)
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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

−8.481801538878562520553309515464, −7.57406462935181192988029021797, −7.02403587128247696597215148409, −6.19327104911231780019891983803, −5.41729196491841690040502753804, −3.88567481688647077948763312627, −2.94915740785686815555861255643, −2.04731694152009267678160814725, −1.54992335423290495248639718259, 0, 1.54992335423290495248639718259, 2.04731694152009267678160814725, 2.94915740785686815555861255643, 3.88567481688647077948763312627, 5.41729196491841690040502753804, 6.19327104911231780019891983803, 7.02403587128247696597215148409, 7.57406462935181192988029021797, 8.481801538878562520553309515464

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