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Properties

Label 2-525-5.4-c3-0-21
Degree $2$
Conductor $525$
Sign $-0.447 + 0.894i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.56i·2-s + 3i·3-s − 22.9·4-s + 16.6·6-s − 7i·7-s + 83.0i·8-s − 9·9-s − 33.6·11-s − 68.7i·12-s + 38.3i·13-s − 38.9·14-s + 278.·16-s + 65.7i·17-s + 50.0i·18-s − 33.3·19-s + ⋯
L(s)  = 1  − 1.96i·2-s + 0.577i·3-s − 2.86·4-s + 1.13·6-s − 0.377i·7-s + 3.66i·8-s − 0.333·9-s − 0.921·11-s − 1.65i·12-s + 0.818i·13-s − 0.743·14-s + 4.34·16-s + 0.937i·17-s + 0.655i·18-s − 0.403·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.199515006\)
\(L(\frac12)\) \(\approx\) \(1.199515006\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 + 5.56iT - 8T^{2} \)
11 \( 1 + 33.6T + 1.33e3T^{2} \)
13 \( 1 - 38.3iT - 2.19e3T^{2} \)
17 \( 1 - 65.7iT - 4.91e3T^{2} \)
19 \( 1 + 33.3T + 6.85e3T^{2} \)
23 \( 1 + 207. iT - 1.21e4T^{2} \)
29 \( 1 - 189.T + 2.43e4T^{2} \)
31 \( 1 - 202.T + 2.97e4T^{2} \)
37 \( 1 + 16.5iT - 5.06e4T^{2} \)
41 \( 1 - 388.T + 6.89e4T^{2} \)
43 \( 1 + 41.8iT - 7.95e4T^{2} \)
47 \( 1 - 368. iT - 1.03e5T^{2} \)
53 \( 1 + 458. iT - 1.48e5T^{2} \)
59 \( 1 + 256.T + 2.05e5T^{2} \)
61 \( 1 + 123.T + 2.26e5T^{2} \)
67 \( 1 + 336. iT - 3.00e5T^{2} \)
71 \( 1 + 453.T + 3.57e5T^{2} \)
73 \( 1 + 22.0iT - 3.89e5T^{2} \)
79 \( 1 + 385.T + 4.93e5T^{2} \)
83 \( 1 + 23.7iT - 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 - 51.9iT - 9.12e5T^{2} \)
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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

−10.57783951848438907364267442200, −9.667336167564547339334832883244, −8.709413961105646030879685693787, −8.074785850660182135705636226784, −6.15525575776431497919313838894, −4.66941688259181731099467194833, −4.33976635885024190730365513303, −3.06848006179196917714893076312, −2.14909256877676162479561497997, −0.63709064877902385948879802118, 0.70478516639309608583575104685, 3.04603317802708108103950266014, 4.63184679413971523783074659270, 5.49273035172931631603560251432, 6.14540574582087829421103123452, 7.25538543734385908712802542621, 7.78515920135462666314132280459, 8.547048423722986466378371270511, 9.451167079780116943424831311369, 10.36420872987837045882558384445

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