Properties

Label 2-525-35.27-c1-0-0
Degree $2$
Conductor $525$
Sign $0.00240 - 0.999i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.00240 - 0.999i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.00240 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0945153 + 0.0942883i\)
\(L(\frac12)\) \(\approx\) \(0.0945153 + 0.0942883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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