Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·2-s + 0.368·3-s + 4.58·4-s + 3.51·5-s − 0.945·6-s + 1.16·7-s − 6.63·8-s − 2.86·9-s − 9.02·10-s − 0.924·11-s + 1.68·12-s + 13-s − 2.98·14-s + 1.29·15-s + 7.84·16-s + 5.92·17-s + 7.34·18-s + 0.924·19-s + 16.1·20-s + 0.428·21-s + 2.37·22-s + 2.41·23-s − 2.44·24-s + 7.37·25-s − 2.56·26-s − 2.16·27-s + 5.32·28-s + ⋯
L(s)  = 1  − 1.81·2-s + 0.212·3-s + 2.29·4-s + 1.57·5-s − 0.386·6-s + 0.439·7-s − 2.34·8-s − 0.954·9-s − 2.85·10-s − 0.278·11-s + 0.487·12-s + 0.277·13-s − 0.796·14-s + 0.334·15-s + 1.96·16-s + 1.43·17-s + 1.73·18-s + 0.212·19-s + 3.60·20-s + 0.0934·21-s + 0.505·22-s + 0.504·23-s − 0.498·24-s + 1.47·25-s − 0.503·26-s − 0.415·27-s + 1.00·28-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 \( 1 + 2.56T + 2T^{2} \)
3 \( 1 - 0.368T + 3T^{2} \)
5 \( 1 - 3.51T + 5T^{2} \)
7 \( 1 - 1.16T + 7T^{2} \)
11 \( 1 + 0.924T + 11T^{2} \)
17 \( 1 - 5.92T + 17T^{2} \)
19 \( 1 - 0.924T + 19T^{2} \)
23 \( 1 - 2.41T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 + 0.769T + 31T^{2} \)
37 \( 1 + 6.08T + 37T^{2} \)
41 \( 1 + 4.97T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 4.56T + 59T^{2} \)
61 \( 1 - 0.707T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 3.66T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + 1.72T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 5.41T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.923338021049272422226811299633, −6.84657219520447301230148685017, −6.39953074885191291687918852464, −5.45859424709484896602002114017, −5.21542725526199072082509158816, −3.31914163648594043411299904687, −2.80451344198087213980166130944, −1.68940228670234899135804153852, −1.49813601132598597557717272720, 0, 1.49813601132598597557717272720, 1.68940228670234899135804153852, 2.80451344198087213980166130944, 3.31914163648594043411299904687, 5.21542725526199072082509158816, 5.45859424709484896602002114017, 6.39953074885191291687918852464, 6.84657219520447301230148685017, 7.923338021049272422226811299633

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