Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $0.00240 + 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.540 + 0.540i)2-s + (−0.707 + 0.707i)3-s + 1.41i·4-s − 0.763i·6-s + (−2.57 + 0.614i)7-s + (−1.84 − 1.84i)8-s − 1.00i·9-s − 3.85·11-s + (−1.00 − 1.00i)12-s + (3.66 − 3.66i)13-s + (1.05 − 1.72i)14-s − 0.839·16-s + (−1.49 − 1.49i)17-s + (0.540 + 0.540i)18-s + 0.0697·19-s + ⋯
L(s)  = 1  + (−0.381 + 0.381i)2-s + (−0.408 + 0.408i)3-s + 0.708i·4-s − 0.311i·6-s + (−0.972 + 0.232i)7-s + (−0.652 − 0.652i)8-s − 0.333i·9-s − 1.16·11-s + (−0.289 − 0.289i)12-s + (1.01 − 1.01i)13-s + (0.282 − 0.460i)14-s − 0.209·16-s + (−0.361 − 0.361i)17-s + (0.127 + 0.127i)18-s + 0.0160·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00240 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00240 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.00240 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{525} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 0.00240 + 0.999i)\)
\(L(1)\)  \(\approx\)  \(0.0945153 - 0.0942883i\)
\(L(\frac12)\)  \(\approx\)  \(0.0945153 - 0.0942883i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (2.57 - 0.614i)T \)
good2 \( 1 + (0.540 - 0.540i)T - 2iT^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + (-3.66 + 3.66i)T - 13iT^{2} \)
17 \( 1 + (1.49 + 1.49i)T + 17iT^{2} \)
19 \( 1 - 0.0697T + 19T^{2} \)
23 \( 1 + (-0.534 - 0.534i)T + 23iT^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + (6.18 - 6.18i)T - 37iT^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (-2.77 - 2.77i)T + 43iT^{2} \)
47 \( 1 + (5.49 + 5.49i)T + 47iT^{2} \)
53 \( 1 + (6.13 + 6.13i)T + 53iT^{2} \)
59 \( 1 + 6.97T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (0.416 - 0.416i)T - 67iT^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (9.55 - 9.55i)T - 73iT^{2} \)
79 \( 1 - 9.86iT - 79T^{2} \)
83 \( 1 + (1.63 - 1.63i)T - 83iT^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + (6.85 + 6.85i)T + 97iT^{2} \)
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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

−10.49405432976239070167491249784, −9.754034626271866192253784947530, −8.762692546661284601434284611268, −8.060300485112853335505769267237, −6.99038096472629603635996150395, −6.12133936915067993537634470032, −5.13704872719110586329851820851, −3.65947128566355129490678540824, −2.87653550663845436799957445531, −0.091484714486376494649915062287, 1.55966710606836978890085568676, 2.92308900295892112551831417814, 4.48527303214564145669980829426, 5.77926733881250603969427420240, 6.35354386366421143571987867537, 7.38572650932123473648041869154, 8.656288225147340428454327253315, 9.385553598145505034819636047197, 10.42446425855002691250333464187, 10.85139984206772436473650214453

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