Properties

Label 2-4017-1.1-c1-0-51
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  + 0.921·2-s + 3-s − 1.15·4-s − 1.44·5-s + 0.921·6-s − 0.134·7-s − 2.90·8-s + 9-s − 1.33·10-s + 3.30·11-s − 1.15·12-s + 13-s − 0.123·14-s − 1.44·15-s − 0.370·16-s − 0.860·17-s + 0.921·18-s − 3.82·19-s + 1.66·20-s − 0.134·21-s + 3.04·22-s + 8.84·23-s − 2.90·24-s − 2.90·25-s + 0.921·26-s + 27-s + 0.154·28-s + ⋯
L(s)  = 1  + 0.651·2-s + 0.577·3-s − 0.575·4-s − 0.646·5-s + 0.376·6-s − 0.0508·7-s − 1.02·8-s + 0.333·9-s − 0.421·10-s + 0.997·11-s − 0.332·12-s + 0.277·13-s − 0.0331·14-s − 0.373·15-s − 0.0926·16-s − 0.208·17-s + 0.217·18-s − 0.877·19-s + 0.372·20-s − 0.0293·21-s + 0.649·22-s + 1.84·23-s − 0.592·24-s − 0.581·25-s + 0.180·26-s + 0.192·27-s + 0.0292·28-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

\(L(1)\) \(\approx\) \(2.299576379\)
\(L(\frac12)\) \(\approx\) \(2.299576379\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.921T + 2T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + 0.134T + 7T^{2} \)
11 \( 1 - 3.30T + 11T^{2} \)
17 \( 1 + 0.860T + 17T^{2} \)
19 \( 1 + 3.82T + 19T^{2} \)
23 \( 1 - 8.84T + 23T^{2} \)
29 \( 1 - 1.16T + 29T^{2} \)
31 \( 1 + 8.77T + 31T^{2} \)
37 \( 1 + 4.13T + 37T^{2} \)
41 \( 1 - 8.57T + 41T^{2} \)
43 \( 1 + 0.325T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 1.45T + 53T^{2} \)
59 \( 1 + 1.43T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 6.73T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 4.94T + 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−8.632179438459780720060574079872, −7.73413963889326362926644919541, −6.95421458131001825709351658436, −6.22219875171752222760984510876, −5.29052712023423817117927342232, −4.47710611781428676088100878811, −3.80699384970828411849664348189, −3.35740917436074824716599939813, −2.17620747928529158192460529865, −0.77347616369165406749839573586, 0.77347616369165406749839573586, 2.17620747928529158192460529865, 3.35740917436074824716599939813, 3.80699384970828411849664348189, 4.47710611781428676088100878811, 5.29052712023423817117927342232, 6.22219875171752222760984510876, 6.95421458131001825709351658436, 7.73413963889326362926644919541, 8.632179438459780720060574079872

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