Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 53 $
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  − 3.25·3-s + 3.01·5-s + 1.05·7-s + 7.62·9-s + 0.331·11-s + 5.94·13-s − 9.83·15-s − 5.88·17-s − 19-s − 3.43·21-s − 7.02·23-s + 4.11·25-s − 15.0·27-s − 0.630·29-s + 1.75·31-s − 1.08·33-s + 3.18·35-s − 8.41·37-s − 19.3·39-s − 3.43·41-s − 12.5·43-s + 23.0·45-s − 11.6·47-s − 5.88·49-s + 19.1·51-s + 53-s + 1.00·55-s + ⋯
L(s)  = 1  − 1.88·3-s + 1.35·5-s + 0.398·7-s + 2.54·9-s + 0.0999·11-s + 1.65·13-s − 2.54·15-s − 1.42·17-s − 0.229·19-s − 0.750·21-s − 1.46·23-s + 0.822·25-s − 2.90·27-s − 0.117·29-s + 0.314·31-s − 0.188·33-s + 0.538·35-s − 1.38·37-s − 3.10·39-s − 0.535·41-s − 1.90·43-s + 3.43·45-s − 1.70·47-s − 0.841·49-s + 2.68·51-s + 0.137·53-s + 0.134·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4028\)    =    \(2^{2} \cdot 19 \cdot 53\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4028} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4028,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;53\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 + T \)
53 \( 1 - T \)
good3 \( 1 + 3.25T + 3T^{2} \)
5 \( 1 - 3.01T + 5T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 - 0.331T + 11T^{2} \)
13 \( 1 - 5.94T + 13T^{2} \)
17 \( 1 + 5.88T + 17T^{2} \)
23 \( 1 + 7.02T + 23T^{2} \)
29 \( 1 + 0.630T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 8.41T + 37T^{2} \)
41 \( 1 + 3.43T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
59 \( 1 - 5.04T + 59T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 + 9.48T + 67T^{2} \)
71 \( 1 - 3.45T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 1.29T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 - 0.759T + 89T^{2} \)
97 \( 1 + 7.99T + 97T^{2} \)
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

−8.138956341732645480190678003927, −6.68544450547090572804448538708, −6.58071044107221361578519154906, −5.89377999298108925965612818424, −5.26869370937162429175098214545, −4.58025819682148579971883761272, −3.67806271587458605571926056262, −1.89447757224690494296071058385, −1.49039218470837446447822748832, 0, 1.49039218470837446447822748832, 1.89447757224690494296071058385, 3.67806271587458605571926056262, 4.58025819682148579971883761272, 5.26869370937162429175098214545, 5.89377999298108925965612818424, 6.58071044107221361578519154906, 6.68544450547090572804448538708, 8.138956341732645480190678003927

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