Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.792 - 0.609i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.800 + 0.800i)2-s + (1.34 + 1.09i)3-s − 0.718i·4-s + (0.199 + 1.95i)6-s + (0.707 − 0.707i)7-s + (2.17 − 2.17i)8-s + (0.606 + 2.93i)9-s − 5.20i·11-s + (0.785 − 0.964i)12-s + (3.24 + 3.24i)13-s + 1.13·14-s + 2.04·16-s + (0.844 + 0.844i)17-s + (−1.86 + 2.83i)18-s + 1.32i·19-s + ⋯
L(s)  = 1  + (0.566 + 0.566i)2-s + (0.775 + 0.631i)3-s − 0.359i·4-s + (0.0813 + 0.796i)6-s + (0.267 − 0.267i)7-s + (0.769 − 0.769i)8-s + (0.202 + 0.979i)9-s − 1.56i·11-s + (0.226 − 0.278i)12-s + (0.900 + 0.900i)13-s + 0.302·14-s + 0.511·16-s + (0.204 + 0.204i)17-s + (−0.439 + 0.668i)18-s + 0.302i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.792 - 0.609i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (218, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.792 - 0.609i)\)
\(L(1)\)  \(\approx\)  \(2.47533 + 0.841445i\)
\(L(\frac12)\)  \(\approx\)  \(2.47533 + 0.841445i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.09i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-0.800 - 0.800i)T + 2iT^{2} \)
11 \( 1 + 5.20iT - 11T^{2} \)
13 \( 1 + (-3.24 - 3.24i)T + 13iT^{2} \)
17 \( 1 + (-0.844 - 0.844i)T + 17iT^{2} \)
19 \( 1 - 1.32iT - 19T^{2} \)
23 \( 1 + (5.62 - 5.62i)T - 23iT^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + (-1.71 + 1.71i)T - 37iT^{2} \)
41 \( 1 + 1.82iT - 41T^{2} \)
43 \( 1 + (-0.281 - 0.281i)T + 43iT^{2} \)
47 \( 1 + (3.39 + 3.39i)T + 47iT^{2} \)
53 \( 1 + (-3.51 + 3.51i)T - 53iT^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (7.92 - 7.92i)T - 67iT^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + (-1.33 - 1.33i)T + 73iT^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 + (5.46 - 5.46i)T - 83iT^{2} \)
89 \( 1 - 9.43T + 89T^{2} \)
97 \( 1 + (-3.06 + 3.06i)T - 97iT^{2} \)
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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

−10.89116010434743820333002973926, −10.04175385476695537685186061151, −9.123110644047595747570004540142, −8.275340144163189013140717974131, −7.36023282869682159505065473391, −6.09391864537988335359899365816, −5.43877277382276287274814859269, −4.11677888459488984507473618273, −3.55177002571541868316778655178, −1.64503526501902550920110844079, 1.76188999677948735560506003032, 2.70454402142320672237116043978, 3.81377150413563260285002952699, 4.78148784028142059235618683352, 6.18735153748007605950658840062, 7.41997635064764723211050645721, 7.973504559639976286021565672161, 8.860934786305970789250653330993, 9.931627239946769293042302259917, 10.93053947244279752079617786342

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