Properties

Label 2-4015-1.1-c1-0-172
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  + 0.416·2-s + 1.40·3-s − 1.82·4-s − 5-s + 0.584·6-s − 1.52·7-s − 1.59·8-s − 1.03·9-s − 0.416·10-s + 11-s − 2.56·12-s + 2.05·13-s − 0.637·14-s − 1.40·15-s + 2.98·16-s + 6.91·17-s − 0.430·18-s − 1.02·19-s + 1.82·20-s − 2.14·21-s + 0.416·22-s − 0.555·23-s − 2.23·24-s + 25-s + 0.858·26-s − 5.65·27-s + 2.79·28-s + ⋯
L(s)  = 1  + 0.294·2-s + 0.809·3-s − 0.913·4-s − 0.447·5-s + 0.238·6-s − 0.577·7-s − 0.563·8-s − 0.344·9-s − 0.131·10-s + 0.301·11-s − 0.739·12-s + 0.571·13-s − 0.170·14-s − 0.362·15-s + 0.747·16-s + 1.67·17-s − 0.101·18-s − 0.235·19-s + 0.408·20-s − 0.468·21-s + 0.0888·22-s − 0.115·23-s − 0.456·24-s + 0.200·25-s + 0.168·26-s − 1.08·27-s + 0.527·28-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 \( 1 - 0.416T + 2T^{2} \)
3 \( 1 - 1.40T + 3T^{2} \)
7 \( 1 + 1.52T + 7T^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
17 \( 1 - 6.91T + 17T^{2} \)
19 \( 1 + 1.02T + 19T^{2} \)
23 \( 1 + 0.555T + 23T^{2} \)
29 \( 1 - 8.69T + 29T^{2} \)
31 \( 1 + 9.18T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 + 7.10T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 9.47T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 3.95T + 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + 3.25T + 71T^{2} \)
79 \( 1 + 8.38T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + 6.12T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−8.113120181494657202415807736591, −7.64136331426232090437547194929, −6.53827620772107732819069751294, −5.75610699305362585298635539614, −5.05181073360994030934947811341, −3.99087688824791355907845857784, −3.43405542757590417138194564453, −2.93145006291258470646122824306, −1.38819387810353001661151637366, 0, 1.38819387810353001661151637366, 2.93145006291258470646122824306, 3.43405542757590417138194564453, 3.99087688824791355907845857784, 5.05181073360994030934947811341, 5.75610699305362585298635539614, 6.53827620772107732819069751294, 7.64136331426232090437547194929, 8.113120181494657202415807736591

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