Properties

Degree 2
Conductor $ 5 \cdot 7 \cdot 229 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.431·2-s + 3.18·3-s − 1.81·4-s − 5-s − 1.37·6-s − 7-s + 1.64·8-s + 7.15·9-s + 0.431·10-s + 3.80·11-s − 5.78·12-s − 2.66·13-s + 0.431·14-s − 3.18·15-s + 2.91·16-s + 4.14·17-s − 3.08·18-s + 0.188·19-s + 1.81·20-s − 3.18·21-s − 1.64·22-s − 5.11·23-s + 5.24·24-s + 25-s + 1.14·26-s + 13.2·27-s + 1.81·28-s + ⋯
L(s)  = 1  − 0.304·2-s + 1.84·3-s − 0.906·4-s − 0.447·5-s − 0.561·6-s − 0.377·7-s + 0.581·8-s + 2.38·9-s + 0.136·10-s + 1.14·11-s − 1.66·12-s − 0.737·13-s + 0.115·14-s − 0.822·15-s + 0.729·16-s + 1.00·17-s − 0.727·18-s + 0.0432·19-s + 0.405·20-s − 0.695·21-s − 0.349·22-s − 1.06·23-s + 1.07·24-s + 0.200·25-s + 0.225·26-s + 2.55·27-s + 0.342·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8015\)    =    \(5 \cdot 7 \cdot 229\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.830494872$
$L(\frac12)$  $\approx$  $2.830494872$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7,\;229\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7,\;229\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 + 0.431T + 2T^{2} \)
3 \( 1 - 3.18T + 3T^{2} \)
11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 + 2.66T + 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 - 0.188T + 19T^{2} \)
23 \( 1 + 5.11T + 23T^{2} \)
29 \( 1 - 6.66T + 29T^{2} \)
31 \( 1 + 3.15T + 31T^{2} \)
37 \( 1 + 5.04T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 8.17T + 43T^{2} \)
47 \( 1 + 0.591T + 47T^{2} \)
53 \( 1 + 4.28T + 53T^{2} \)
59 \( 1 - 8.59T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 - 7.44T + 71T^{2} \)
73 \( 1 - 2.33T + 73T^{2} \)
79 \( 1 - 7.64T + 79T^{2} \)
83 \( 1 - 5.54T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−7.87047649607883129875799059778, −7.56241481955241968777504956649, −6.78080579099902082731505880854, −5.75597593189770451594215798623, −4.60036650464214973622397067061, −4.11526963858887196448197985432, −3.52252108800731269422506070343, −2.83453333908905959206013743654, −1.79456752755153655623899245737, −0.838848795876563996131958940493, 0.838848795876563996131958940493, 1.79456752755153655623899245737, 2.83453333908905959206013743654, 3.52252108800731269422506070343, 4.11526963858887196448197985432, 4.60036650464214973622397067061, 5.75597593189770451594215798623, 6.78080579099902082731505880854, 7.56241481955241968777504956649, 7.87047649607883129875799059778

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