Properties

Degree 2
Conductor 1931
Sign $-1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 5-s + 7-s − 9-s + 1.41i·10-s + 11-s − 1.41i·14-s − 0.999·16-s − 1.41i·17-s + 1.41i·18-s + 1.00·20-s − 1.41i·22-s − 23-s − 1.00·28-s − 1.41i·29-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 5-s + 7-s − 9-s + 1.41i·10-s + 11-s − 1.41i·14-s − 0.999·16-s − 1.41i·17-s + 1.41i·18-s + 1.00·20-s − 1.41i·22-s − 23-s − 1.00·28-s − 1.41i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1931\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{1931} (1930, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1931,\ (\ :0),\ -1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.8649243734\)
\(L(\frac12)\)  \(\approx\)  \(0.8649243734\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 1931$,\(F_p(T)\) is a polynomial of degree 2. If $p = 1931$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad1931 \( 1 + T \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.302509833227012902750251345152, −8.207548641401306159956266739135, −7.79647987360422128281152910843, −6.66290335780759212989203062536, −5.58797453066732782513570994341, −4.37701414726984926262682226186, −4.01980205705184025465392034462, −2.90702303416611842834557589692, −2.05053721925341121864616052503, −0.64136295946191236907305012765, 1.73721170950184380171242786625, 3.38007219860893007310652654470, 4.30803944960578103971401159942, 5.05917423423338770058612401516, 6.02036187512070653734499900733, 6.55664024565111314950222763948, 7.62066731358320132202703376964, 8.068679183494150857034889085580, 8.595002379770916575626299736610, 9.271962465978717242274382304893

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