Properties

Label 2-4017-1.1-c1-0-37
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  − 2.39·2-s + 3-s + 3.75·4-s − 1.39·5-s − 2.39·6-s − 0.807·7-s − 4.21·8-s + 9-s + 3.33·10-s + 2.13·11-s + 3.75·12-s + 13-s + 1.93·14-s − 1.39·15-s + 2.60·16-s + 1.24·17-s − 2.39·18-s + 3.51·19-s − 5.22·20-s − 0.807·21-s − 5.13·22-s − 5.94·23-s − 4.21·24-s − 3.06·25-s − 2.39·26-s + 27-s − 3.03·28-s + ⋯
L(s)  = 1  − 1.69·2-s + 0.577·3-s + 1.87·4-s − 0.622·5-s − 0.979·6-s − 0.305·7-s − 1.49·8-s + 0.333·9-s + 1.05·10-s + 0.644·11-s + 1.08·12-s + 0.277·13-s + 0.518·14-s − 0.359·15-s + 0.650·16-s + 0.302·17-s − 0.565·18-s + 0.807·19-s − 1.16·20-s − 0.176·21-s − 1.09·22-s − 1.23·23-s − 0.860·24-s − 0.612·25-s − 0.470·26-s + 0.192·27-s − 0.573·28-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

\(L(1)\) \(\approx\) \(0.8567789185\)
\(L(\frac12)\) \(\approx\) \(0.8567789185\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.39T + 2T^{2} \)
5 \( 1 + 1.39T + 5T^{2} \)
7 \( 1 + 0.807T + 7T^{2} \)
11 \( 1 - 2.13T + 11T^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 + 5.94T + 23T^{2} \)
29 \( 1 + 1.89T + 29T^{2} \)
31 \( 1 - 4.81T + 31T^{2} \)
37 \( 1 - 7.11T + 37T^{2} \)
41 \( 1 + 0.143T + 41T^{2} \)
43 \( 1 - 9.57T + 43T^{2} \)
47 \( 1 - 1.05T + 47T^{2} \)
53 \( 1 + 2.69T + 53T^{2} \)
59 \( 1 + 4.33T + 59T^{2} \)
61 \( 1 - 5.95T + 61T^{2} \)
67 \( 1 + 0.992T + 67T^{2} \)
71 \( 1 + 5.26T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + 4.53T + 79T^{2} \)
83 \( 1 - 7.02T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−8.426579383956645269482762338528, −7.80372635474325254613237842255, −7.48262104503986083023111247991, −6.53210821202328673573789624385, −5.88879798973256346810038921340, −4.42296826015305458317802314562, −3.63276674342619978503204743134, −2.68962174859401015072683445643, −1.68299245306879170033808459341, −0.68246221984669281339794167555, 0.68246221984669281339794167555, 1.68299245306879170033808459341, 2.68962174859401015072683445643, 3.63276674342619978503204743134, 4.42296826015305458317802314562, 5.88879798973256346810038921340, 6.53210821202328673573789624385, 7.48262104503986083023111247991, 7.80372635474325254613237842255, 8.426579383956645269482762338528

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