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  − 2.18·2-s − 3-s + 2.76·4-s + 2.14·5-s + 2.18·6-s + 0.603·7-s − 1.67·8-s + 9-s − 4.68·10-s − 1.05·11-s − 2.76·12-s + 13-s − 1.31·14-s − 2.14·15-s − 1.88·16-s + 5.27·17-s − 2.18·18-s + 6.92·19-s + 5.93·20-s − 0.603·21-s + 2.30·22-s − 3.16·23-s + 1.67·24-s − 0.388·25-s − 2.18·26-s − 27-s + 1.66·28-s + ⋯
L(s)  = 1  − 1.54·2-s − 0.577·3-s + 1.38·4-s + 0.960·5-s + 0.891·6-s + 0.228·7-s − 0.590·8-s + 0.333·9-s − 1.48·10-s − 0.318·11-s − 0.798·12-s + 0.277·13-s − 0.351·14-s − 0.554·15-s − 0.470·16-s + 1.27·17-s − 0.514·18-s + 1.58·19-s + 1.32·20-s − 0.131·21-s + 0.491·22-s − 0.659·23-s + 0.341·24-s − 0.0776·25-s − 0.428·26-s − 0.192·27-s + 0.315·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9720034632$
$L(\frac12)$  $\approx$  $0.9720034632$
$L(\frac{3}{2})$   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 \( 1 + 2.18T + 2T^{2} \)
5 \( 1 - 2.14T + 5T^{2} \)
7 \( 1 - 0.603T + 7T^{2} \)
11 \( 1 + 1.05T + 11T^{2} \)
17 \( 1 - 5.27T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 - 6.31T + 29T^{2} \)
31 \( 1 - 1.92T + 31T^{2} \)
37 \( 1 + 7.58T + 37T^{2} \)
41 \( 1 + 6.70T + 41T^{2} \)
43 \( 1 - 3.27T + 43T^{2} \)
47 \( 1 - 0.617T + 47T^{2} \)
53 \( 1 - 14.2T + 53T^{2} \)
59 \( 1 - 8.58T + 59T^{2} \)
61 \( 1 - 2.25T + 61T^{2} \)
67 \( 1 - 6.67T + 67T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 3.02T + 79T^{2} \)
83 \( 1 - 6.20T + 83T^{2} \)
89 \( 1 + 6.13T + 89T^{2} \)
97 \( 1 + 5.90T + 97T^{2} \)
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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

−8.404145682268107380572637172813, −7.909877792815525054188617156978, −7.10496011151685314491173461912, −6.45464730194188114932481738229, −5.51194394493893713959152949990, −5.10080611142126338352655379539, −3.68344782128412685700190729494, −2.50691957720305997574906150559, −1.53724159447549874589928213510, −0.800693965236434549622211149522, 0.800693965236434549622211149522, 1.53724159447549874589928213510, 2.50691957720305997574906150559, 3.68344782128412685700190729494, 5.10080611142126338352655379539, 5.51194394493893713959152949990, 6.45464730194188114932481738229, 7.10496011151685314491173461912, 7.909877792815525054188617156978, 8.404145682268107380572637172813

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