Properties

Label 2-3525-705.563-c0-0-9
Degree $2$
Conductor $3525$
Sign $-0.971 + 0.237i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.14 + 1.14i)2-s + (0.453 + 0.891i)3-s + 1.61i·4-s + (−0.5 + 1.53i)6-s + (−0.831 + 0.831i)7-s + (−0.707 + 0.707i)8-s + (−0.587 + 0.809i)9-s + (−1.44 + 0.734i)12-s − 1.90·14-s + (−0.437 − 0.437i)17-s + (−1.59 + 0.253i)18-s + (−1.11 − 0.363i)21-s + (−0.951 − 0.309i)24-s + (−0.987 − 0.156i)27-s + (−1.34 − 1.34i)28-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)2-s + (0.453 + 0.891i)3-s + 1.61i·4-s + (−0.5 + 1.53i)6-s + (−0.831 + 0.831i)7-s + (−0.707 + 0.707i)8-s + (−0.587 + 0.809i)9-s + (−1.44 + 0.734i)12-s − 1.90·14-s + (−0.437 − 0.437i)17-s + (−1.59 + 0.253i)18-s + (−1.11 − 0.363i)21-s + (−0.951 − 0.309i)24-s + (−0.987 − 0.156i)27-s + (−1.34 − 1.34i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.971 + 0.237i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.971 + 0.237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.308445055\)
\(L(\frac12)\) \(\approx\) \(2.308445055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.453 - 0.891i)T \)
5 \( 1 \)
47 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
7 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
59 \( 1 - 1.17T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.90iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 0.618iT - T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
89 \( 1 - 1.17T + T^{2} \)
97 \( 1 + (-0.831 + 0.831i)T - iT^{2} \)
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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

−8.857177301216428591380640365090, −8.460692568687907458274496096683, −7.49402217785291230043236833052, −6.73396706419759330861348535979, −6.03897518826910418097283019702, −5.29935116511432646846972375414, −4.77644789880642852172413310944, −3.83178410998421852210960180882, −3.21757767265092281642443479729, −2.35722145502626766949872242041, 0.859706433913556267121193979715, 2.01092967131162696508748563037, 2.71347841908458053413900050911, 3.76953863564185260987855469048, 3.93148000053939637483333417609, 5.26699849941063075741416748531, 5.97509227735655964248774554838, 6.86084015861444980389600055028, 7.35410565724013852656512009252, 8.435772965644911898294816277516

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