Properties

Label 2-4017-1.1-c1-0-9
Degree $2$
Conductor $4017$
Sign $1$
Analytic cond. $32.0759$
Root an. cond. $5.66355$
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  − 2.42·2-s − 3-s + 3.88·4-s − 0.736·5-s + 2.42·6-s − 2.40·7-s − 4.56·8-s + 9-s + 1.78·10-s − 1.03·11-s − 3.88·12-s − 13-s + 5.83·14-s + 0.736·15-s + 3.30·16-s − 2.49·17-s − 2.42·18-s + 0.457·19-s − 2.85·20-s + 2.40·21-s + 2.50·22-s + 7.34·23-s + 4.56·24-s − 4.45·25-s + 2.42·26-s − 27-s − 9.33·28-s + ⋯
L(s)  = 1  − 1.71·2-s − 0.577·3-s + 1.94·4-s − 0.329·5-s + 0.990·6-s − 0.909·7-s − 1.61·8-s + 0.333·9-s + 0.565·10-s − 0.311·11-s − 1.12·12-s − 0.277·13-s + 1.55·14-s + 0.190·15-s + 0.825·16-s − 0.604·17-s − 0.571·18-s + 0.104·19-s − 0.639·20-s + 0.525·21-s + 0.533·22-s + 1.53·23-s + 0.931·24-s − 0.891·25-s + 0.475·26-s − 0.192·27-s − 1.76·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $1$
Analytic conductor: \(32.0759\)
Root analytic conductor: \(5.66355\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4017,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2236977103\)
\(L(\frac12)\) \(\approx\) \(0.2236977103\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 - T \)
good2 \( 1 + 2.42T + 2T^{2} \)
5 \( 1 + 0.736T + 5T^{2} \)
7 \( 1 + 2.40T + 7T^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
17 \( 1 + 2.49T + 17T^{2} \)
19 \( 1 - 0.457T + 19T^{2} \)
23 \( 1 - 7.34T + 23T^{2} \)
29 \( 1 - 4.43T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 + 9.29T + 41T^{2} \)
43 \( 1 - 6.10T + 43T^{2} \)
47 \( 1 - 7.85T + 47T^{2} \)
53 \( 1 - 2.17T + 53T^{2} \)
59 \( 1 + 9.64T + 59T^{2} \)
61 \( 1 - 7.37T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 - 2.29T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 9.92T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + 5.44T + 97T^{2} \)
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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

−8.719917461536242592379863549957, −7.63207851346957339111250635200, −7.17058553053368027905563477567, −6.61244979239279409373683041675, −5.75567547102475977481898285224, −4.81224937408784882365334295382, −3.61004141914140521299015072171, −2.65578948271441796967478650382, −1.58550839576594511881269813466, −0.36050079290433118173383664741, 0.36050079290433118173383664741, 1.58550839576594511881269813466, 2.65578948271441796967478650382, 3.61004141914140521299015072171, 4.81224937408784882365334295382, 5.75567547102475977481898285224, 6.61244979239279409373683041675, 7.17058553053368027905563477567, 7.63207851346957339111250635200, 8.719917461536242592379863549957

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