Properties

Label 2-4017-1.1-c1-0-36
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  + 0.543·2-s − 3-s − 1.70·4-s + 2.45·5-s − 0.543·6-s + 0.874·7-s − 2.01·8-s + 9-s + 1.33·10-s − 4.72·11-s + 1.70·12-s − 13-s + 0.474·14-s − 2.45·15-s + 2.31·16-s − 1.13·17-s + 0.543·18-s + 3.87·19-s − 4.18·20-s − 0.874·21-s − 2.56·22-s + 0.738·23-s + 2.01·24-s + 1.01·25-s − 0.543·26-s − 27-s − 1.49·28-s + ⋯
L(s)  = 1  + 0.384·2-s − 0.577·3-s − 0.852·4-s + 1.09·5-s − 0.221·6-s + 0.330·7-s − 0.711·8-s + 0.333·9-s + 0.421·10-s − 1.42·11-s + 0.492·12-s − 0.277·13-s + 0.126·14-s − 0.633·15-s + 0.579·16-s − 0.276·17-s + 0.128·18-s + 0.889·19-s − 0.935·20-s − 0.190·21-s − 0.547·22-s + 0.153·23-s + 0.410·24-s + 0.203·25-s − 0.106·26-s − 0.192·27-s − 0.281·28-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

\(L(1)\) \(\approx\) \(1.519126334\)
\(L(\frac12)\) \(\approx\) \(1.519126334\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.543T + 2T^{2} \)
5 \( 1 - 2.45T + 5T^{2} \)
7 \( 1 - 0.874T + 7T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 3.87T + 19T^{2} \)
23 \( 1 - 0.738T + 23T^{2} \)
29 \( 1 - 0.947T + 29T^{2} \)
31 \( 1 + 3.64T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 + 0.380T + 41T^{2} \)
43 \( 1 + 2.38T + 43T^{2} \)
47 \( 1 - 5.91T + 47T^{2} \)
53 \( 1 - 8.85T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 4.69T + 61T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 - 2.39T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 4.05T + 83T^{2} \)
89 \( 1 - 4.49T + 89T^{2} \)
97 \( 1 - 0.478T + 97T^{2} \)
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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

−8.442776505934406099638538887275, −7.74106107533845517881288639692, −6.85781921133627490685048753047, −5.87825207664370234871578846323, −5.34851328763794990938209224697, −5.01566456274453302726495275185, −4.04114706144003486643156440505, −2.93855011673932536515537040612, −2.02526255186431515218142903976, −0.68208339767828069962693595695, 0.68208339767828069962693595695, 2.02526255186431515218142903976, 2.93855011673932536515537040612, 4.04114706144003486643156440505, 5.01566456274453302726495275185, 5.34851328763794990938209224697, 5.87825207664370234871578846323, 6.85781921133627490685048753047, 7.74106107533845517881288639692, 8.442776505934406099638538887275

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