Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·2-s − 3-s + 4.59·4-s + 0.753·5-s − 2.56·6-s + 4.02·7-s + 6.66·8-s + 9-s + 1.93·10-s − 1.53·11-s − 4.59·12-s + 13-s + 10.3·14-s − 0.753·15-s + 7.91·16-s + 4.91·17-s + 2.56·18-s − 0.0697·19-s + 3.46·20-s − 4.02·21-s − 3.93·22-s + 6.24·23-s − 6.66·24-s − 4.43·25-s + 2.56·26-s − 27-s + 18.4·28-s + ⋯
L(s)  = 1  + 1.81·2-s − 0.577·3-s + 2.29·4-s + 0.337·5-s − 1.04·6-s + 1.52·7-s + 2.35·8-s + 0.333·9-s + 0.611·10-s − 0.462·11-s − 1.32·12-s + 0.277·13-s + 2.76·14-s − 0.194·15-s + 1.97·16-s + 1.19·17-s + 0.605·18-s − 0.0160·19-s + 0.774·20-s − 0.878·21-s − 0.839·22-s + 1.30·23-s − 1.35·24-s − 0.886·25-s + 0.503·26-s − 0.192·27-s + 3.49·28-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

\( d \)  =  \(2\)
\( N \)  =  \(4017\)    =    \(3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.706112490$
$L(\frac12)$  $\approx$  $6.706112490$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good2 \( 1 - 2.56T + 2T^{2} \)
5 \( 1 - 0.753T + 5T^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
11 \( 1 + 1.53T + 11T^{2} \)
17 \( 1 - 4.91T + 17T^{2} \)
19 \( 1 + 0.0697T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 + 1.50T + 29T^{2} \)
31 \( 1 + 5.24T + 31T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 - 0.427T + 41T^{2} \)
43 \( 1 - 8.14T + 43T^{2} \)
47 \( 1 + 4.93T + 47T^{2} \)
53 \( 1 + 0.860T + 53T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 - 8.58T + 61T^{2} \)
67 \( 1 - 8.96T + 67T^{2} \)
71 \( 1 - 6.17T + 71T^{2} \)
73 \( 1 + 0.238T + 73T^{2} \)
79 \( 1 + 8.45T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 - 3.54T + 89T^{2} \)
97 \( 1 - 9.72T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.063539481575566080508856896986, −7.45886782592416181660863424902, −6.77326264512461819140181886133, −5.73504670123333395137565831924, −5.39578979229051038537081934450, −4.89135145424147927784191938400, −4.04124502591185106934773339984, −3.22335007068833617011626393554, −2.10754215083019521533864202741, −1.35451113256913630287748212004, 1.35451113256913630287748212004, 2.10754215083019521533864202741, 3.22335007068833617011626393554, 4.04124502591185106934773339984, 4.89135145424147927784191938400, 5.39578979229051038537081934450, 5.73504670123333395137565831924, 6.77326264512461819140181886133, 7.45886782592416181660863424902, 8.063539481575566080508856896986

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