Properties

Label 2-4015-1.1-c1-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.85·2-s − 2.11·3-s + 1.42·4-s + 5-s − 3.91·6-s − 4.74·7-s − 1.05·8-s + 1.46·9-s + 1.85·10-s − 11-s − 3.01·12-s + 3.70·13-s − 8.78·14-s − 2.11·15-s − 4.81·16-s − 5.22·17-s + 2.71·18-s − 4.97·19-s + 1.42·20-s + 10.0·21-s − 1.85·22-s + 5.98·23-s + 2.23·24-s + 25-s + 6.86·26-s + 3.24·27-s − 6.77·28-s + ⋯
L(s)  = 1  + 1.30·2-s − 1.22·3-s + 0.713·4-s + 0.447·5-s − 1.59·6-s − 1.79·7-s − 0.374·8-s + 0.488·9-s + 0.585·10-s − 0.301·11-s − 0.871·12-s + 1.02·13-s − 2.34·14-s − 0.545·15-s − 1.20·16-s − 1.26·17-s + 0.640·18-s − 1.14·19-s + 0.319·20-s + 2.18·21-s − 0.394·22-s + 1.24·23-s + 0.457·24-s + 0.200·25-s + 1.34·26-s + 0.623·27-s − 1.28·28-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: \(0\)
Selberg data: \((2,\ 4015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.242403171\)
\(L(\frac12)\) \(\approx\) \(1.242403171\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.85T + 2T^{2} \)
3 \( 1 + 2.11T + 3T^{2} \)
7 \( 1 + 4.74T + 7T^{2} \)
13 \( 1 - 3.70T + 13T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 + 4.97T + 19T^{2} \)
23 \( 1 - 5.98T + 23T^{2} \)
29 \( 1 + 1.29T + 29T^{2} \)
31 \( 1 + 1.51T + 31T^{2} \)
37 \( 1 - 0.300T + 37T^{2} \)
41 \( 1 - 8.90T + 41T^{2} \)
43 \( 1 + 1.23T + 43T^{2} \)
47 \( 1 + 4.65T + 47T^{2} \)
53 \( 1 - 1.98T + 53T^{2} \)
59 \( 1 - 5.51T + 59T^{2} \)
61 \( 1 - 0.0688T + 61T^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 + 4.59T + 71T^{2} \)
79 \( 1 - 8.84T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 5.11T + 97T^{2} \)
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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

−8.730953528345050232591491977024, −7.09078493993919336780419211425, −6.41429115774881995466621712972, −6.24108721881835197283148013972, −5.57093872161226439773259881707, −4.77502214834100533223886848952, −3.98833202567009565431957344739, −3.17718823474513944453512853537, −2.34247156037800412349951536298, −0.52408857258537604252559117373, 0.52408857258537604252559117373, 2.34247156037800412349951536298, 3.17718823474513944453512853537, 3.98833202567009565431957344739, 4.77502214834100533223886848952, 5.57093872161226439773259881707, 6.24108721881835197283148013972, 6.41429115774881995466621712972, 7.09078493993919336780419211425, 8.730953528345050232591491977024

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