Properties

Label 2-4015-1.1-c1-0-103
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  − 2.34·2-s + 2.98·3-s + 3.50·4-s − 5-s − 6.99·6-s + 0.286·7-s − 3.53·8-s + 5.88·9-s + 2.34·10-s + 11-s + 10.4·12-s − 0.143·13-s − 0.672·14-s − 2.98·15-s + 1.28·16-s + 5.28·17-s − 13.8·18-s + 5.34·19-s − 3.50·20-s + 0.854·21-s − 2.34·22-s − 0.898·23-s − 10.5·24-s + 25-s + 0.337·26-s + 8.60·27-s + 1.00·28-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.72·3-s + 1.75·4-s − 0.447·5-s − 2.85·6-s + 0.108·7-s − 1.24·8-s + 1.96·9-s + 0.742·10-s + 0.301·11-s + 3.01·12-s − 0.0399·13-s − 0.179·14-s − 0.769·15-s + 0.320·16-s + 1.28·17-s − 3.25·18-s + 1.22·19-s − 0.784·20-s + 0.186·21-s − 0.500·22-s − 0.187·23-s − 2.15·24-s + 0.200·25-s + 0.0662·26-s + 1.65·27-s + 0.189·28-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

\(L(1)\) \(\approx\) \(1.750392559\)
\(L(\frac12)\) \(\approx\) \(1.750392559\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.34T + 2T^{2} \)
3 \( 1 - 2.98T + 3T^{2} \)
7 \( 1 - 0.286T + 7T^{2} \)
13 \( 1 + 0.143T + 13T^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
19 \( 1 - 5.34T + 19T^{2} \)
23 \( 1 + 0.898T + 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 - 0.737T + 37T^{2} \)
41 \( 1 - 8.18T + 41T^{2} \)
43 \( 1 - 2.25T + 43T^{2} \)
47 \( 1 - 0.731T + 47T^{2} \)
53 \( 1 + 2.72T + 53T^{2} \)
59 \( 1 - 9.25T + 59T^{2} \)
61 \( 1 + 2.77T + 61T^{2} \)
67 \( 1 - 7.49T + 67T^{2} \)
71 \( 1 + 6.55T + 71T^{2} \)
79 \( 1 + 6.30T + 79T^{2} \)
83 \( 1 - 2.13T + 83T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−8.396068469899865553266796702822, −7.935929765342046180089029049989, −7.43990410066200298469817233970, −6.91442201238216872190279341035, −5.65261236600141929782248928587, −4.35377135332384490453275664117, −3.42782814586195775884654474335, −2.78738771092642792098772453535, −1.77816303109597716241816891640, −0.953933458840195942403464143175, 0.953933458840195942403464143175, 1.77816303109597716241816891640, 2.78738771092642792098772453535, 3.42782814586195775884654474335, 4.35377135332384490453275664117, 5.65261236600141929782248928587, 6.91442201238216872190279341035, 7.43990410066200298469817233970, 7.935929765342046180089029049989, 8.396068469899865553266796702822

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