Properties

Degree 2
Conductor $ 13 \cdot 617 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.48·2-s − 2.97·3-s + 4.17·4-s + 3.21·5-s + 7.38·6-s + 2.90·7-s − 5.40·8-s + 5.83·9-s − 8.00·10-s − 2.09·11-s − 12.4·12-s + 13-s − 7.21·14-s − 9.56·15-s + 5.07·16-s + 0.183·17-s − 14.4·18-s − 8.32·19-s + 13.4·20-s − 8.62·21-s + 5.20·22-s − 5.93·23-s + 16.0·24-s + 5.36·25-s − 2.48·26-s − 8.41·27-s + 12.1·28-s + ⋯
L(s)  = 1  − 1.75·2-s − 1.71·3-s + 2.08·4-s + 1.43·5-s + 3.01·6-s + 1.09·7-s − 1.91·8-s + 1.94·9-s − 2.52·10-s − 0.632·11-s − 3.58·12-s + 0.277·13-s − 1.92·14-s − 2.47·15-s + 1.26·16-s + 0.0444·17-s − 3.41·18-s − 1.91·19-s + 3.00·20-s − 1.88·21-s + 1.11·22-s − 1.23·23-s + 3.27·24-s + 1.07·25-s − 0.487·26-s − 1.62·27-s + 2.28·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8021\)    =    \(13 \cdot 617\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8021} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8021,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;617\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;617\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 - T \)
617 \( 1 - T \)
good2 \( 1 + 2.48T + 2T^{2} \)
3 \( 1 + 2.97T + 3T^{2} \)
5 \( 1 - 3.21T + 5T^{2} \)
7 \( 1 - 2.90T + 7T^{2} \)
11 \( 1 + 2.09T + 11T^{2} \)
17 \( 1 - 0.183T + 17T^{2} \)
19 \( 1 + 8.32T + 19T^{2} \)
23 \( 1 + 5.93T + 23T^{2} \)
29 \( 1 + 2.88T + 29T^{2} \)
31 \( 1 - 1.57T + 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 - 5.76T + 41T^{2} \)
43 \( 1 + 2.35T + 43T^{2} \)
47 \( 1 + 4.35T + 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 - 4.68T + 59T^{2} \)
61 \( 1 - 9.39T + 61T^{2} \)
67 \( 1 - 1.23T + 67T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 - 5.14T + 73T^{2} \)
79 \( 1 + 0.243T + 79T^{2} \)
83 \( 1 + 1.78T + 83T^{2} \)
89 \( 1 - 1.59T + 89T^{2} \)
97 \( 1 - 9.98T + 97T^{2} \)
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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

−7.52191021099908972899982439509, −6.70265997347473763273075652795, −6.27825364378506975057090147416, −5.62754830419197409772299174203, −5.08097033724160726534150158195, −4.12883406705639274997056412494, −2.18794502821982939806374135779, −1.97179493548145571727840103519, −1.03520433921332721529752824507, 0, 1.03520433921332721529752824507, 1.97179493548145571727840103519, 2.18794502821982939806374135779, 4.12883406705639274997056412494, 5.08097033724160726534150158195, 5.62754830419197409772299174203, 6.27825364378506975057090147416, 6.70265997347473763273075652795, 7.52191021099908972899982439509

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