Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.922 - 0.387i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3.20i·3-s + 1.49i·5-s − 2.89i·7-s − 7.25·9-s − 0.590i·11-s + 6.18·13-s − 4.80·15-s + (3.80 + 1.59i)17-s − 2.41·19-s + 9.27·21-s + 6.41i·23-s + 2.75·25-s − 13.6i·27-s + 6.45i·29-s − 1.17i·31-s + ⋯
L(s)  = 1  + 1.84i·3-s + 0.670i·5-s − 1.09i·7-s − 2.41·9-s − 0.177i·11-s + 1.71·13-s − 1.23·15-s + (0.922 + 0.387i)17-s − 0.554·19-s + 2.02·21-s + 1.33i·23-s + 0.550·25-s − 2.62i·27-s + 1.19i·29-s − 0.210i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.922 - 0.387i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.922 - 0.387i)$
$L(1)$  $\approx$  $1.822066267$
$L(\frac12)$  $\approx$  $1.822066267$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-3.80 - 1.59i)T \)
59 \( 1 - T \)
good3 \( 1 - 3.20iT - 3T^{2} \)
5 \( 1 - 1.49iT - 5T^{2} \)
7 \( 1 + 2.89iT - 7T^{2} \)
11 \( 1 + 0.590iT - 11T^{2} \)
13 \( 1 - 6.18T + 13T^{2} \)
19 \( 1 + 2.41T + 19T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 - 6.45iT - 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + 0.792iT - 37T^{2} \)
41 \( 1 - 3.09iT - 41T^{2} \)
43 \( 1 + 0.984T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 4.09T + 53T^{2} \)
61 \( 1 + 8.04iT - 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 7.20iT - 71T^{2} \)
73 \( 1 + 3.29iT - 73T^{2} \)
79 \( 1 - 10.9iT - 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 + 2.40T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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

−8.908987088239041660009962503905, −8.219229493068584089560466434149, −7.36101565660134464773847401750, −6.36350587138327826473602413252, −5.71887101403394264987909797617, −4.91790470582103613868031723947, −3.82686670715728144335934259348, −3.73597765907413395056413260485, −2.93576095885336349444624563257, −1.20226394680276799487553493956, 0.59737689824590816945473272524, 1.44231828950491129875372243654, 2.32747809326274108049382529979, 3.09874483502615663461284949497, 4.40254765242873356957465209863, 5.56465293251587745357957717454, 5.97779684282057221254416512889, 6.56553110815117313596511363588, 7.46193615365891406248382110406, 8.168767495369541298930964686892

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