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  − 1.97·2-s − 3-s + 1.91·4-s + 2.16·5-s + 1.97·6-s + 5.12·7-s + 0.160·8-s + 9-s − 4.27·10-s + 5.28·11-s − 1.91·12-s + 13-s − 10.1·14-s − 2.16·15-s − 4.15·16-s − 0.0913·17-s − 1.97·18-s − 3.04·19-s + 4.14·20-s − 5.12·21-s − 10.4·22-s + 0.448·23-s − 0.160·24-s − 0.333·25-s − 1.97·26-s − 27-s + 9.83·28-s + ⋯
L(s)  = 1  − 1.39·2-s − 0.577·3-s + 0.959·4-s + 0.966·5-s + 0.808·6-s + 1.93·7-s + 0.0566·8-s + 0.333·9-s − 1.35·10-s + 1.59·11-s − 0.554·12-s + 0.277·13-s − 2.71·14-s − 0.557·15-s − 1.03·16-s − 0.0221·17-s − 0.466·18-s − 0.699·19-s + 0.926·20-s − 1.11·21-s − 2.22·22-s + 0.0934·23-s − 0.0326·24-s − 0.0667·25-s − 0.388·26-s − 0.192·27-s + 1.85·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.416097777$
$L(\frac12)$  $\approx$  $1.416097777$
$L(\frac{3}{2})$   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 \( 1 + 1.97T + 2T^{2} \)
5 \( 1 - 2.16T + 5T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 5.28T + 11T^{2} \)
17 \( 1 + 0.0913T + 17T^{2} \)
19 \( 1 + 3.04T + 19T^{2} \)
23 \( 1 - 0.448T + 23T^{2} \)
29 \( 1 + 2.04T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 - 3.51T + 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 2.08T + 43T^{2} \)
47 \( 1 - 2.87T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 + 4.87T + 59T^{2} \)
61 \( 1 - 9.71T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 3.47T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 5.69T + 97T^{2} \)
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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

−8.631912675465311899256088989259, −7.914714074410932226900712862603, −7.09794958381795615836798020959, −6.39026032799872426529717191756, −5.64951395140154877234232682171, −4.66536380414072730783024961843, −4.14155358842188040550981880504, −2.26864364283584108697697352740, −1.55295356168241304146526126794, −1.00819149862694652931888540769, 1.00819149862694652931888540769, 1.55295356168241304146526126794, 2.26864364283584108697697352740, 4.14155358842188040550981880504, 4.66536380414072730783024961843, 5.64951395140154877234232682171, 6.39026032799872426529717191756, 7.09794958381795615836798020959, 7.914714074410932226900712862603, 8.631912675465311899256088989259

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