Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s − 3-s + 3.82·4-s − 3.41·5-s + 2.41·6-s + 2.82·7-s − 4.41·8-s + 9-s + 8.24·10-s + 1.41·11-s − 3.82·12-s + 13-s − 6.82·14-s + 3.41·15-s + 2.99·16-s + 1.17·17-s − 2.41·18-s + 5.65·19-s − 13.0·20-s − 2.82·21-s − 3.41·22-s − 3.65·23-s + 4.41·24-s + 6.65·25-s − 2.41·26-s − 27-s + 10.8·28-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.577·3-s + 1.91·4-s − 1.52·5-s + 0.985·6-s + 1.06·7-s − 1.56·8-s + 0.333·9-s + 2.60·10-s + 0.426·11-s − 1.10·12-s + 0.277·13-s − 1.82·14-s + 0.881·15-s + 0.749·16-s + 0.284·17-s − 0.569·18-s + 1.29·19-s − 2.92·20-s − 0.617·21-s − 0.727·22-s − 0.762·23-s + 0.901·24-s + 1.33·25-s − 0.473·26-s − 0.192·27-s + 2.04·28-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  =  1
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 - T \)
good2 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 + 3.07T + 37T^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 - 2.82T + 43T^{2} \)
47 \( 1 + 7.07T + 47T^{2} \)
53 \( 1 - 7.65T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 7.41T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 4.82T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−7.87073897975558544031398277480, −7.63801172658218308492118542963, −7.16708748858738772212216473851, −6.08109876301496222359277774549, −5.16156155752868525817998453762, −4.17677202957386955320901384042, −3.39006966306001376982024128932, −1.89850389442155214634578328011, −1.06346747357381321427304379993, 0, 1.06346747357381321427304379993, 1.89850389442155214634578328011, 3.39006966306001376982024128932, 4.17677202957386955320901384042, 5.16156155752868525817998453762, 6.08109876301496222359277774549, 7.16708748858738772212216473851, 7.63801172658218308492118542963, 7.87073897975558544031398277480

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