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Properties

Label 2-3525-705.563-c0-0-22
Degree $2$
Conductor $3525$
Sign $0.765 - 0.643i$
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  + (0.946 + 0.946i)2-s + (0.933 − 0.358i)3-s + 0.790i·4-s + (1.22 + 0.544i)6-s + (−0.294 + 0.294i)7-s + (0.197 − 0.197i)8-s + (0.743 − 0.669i)9-s + (0.283 + 0.738i)12-s − 0.556·14-s + 1.16·16-s + (−0.147 − 0.147i)17-s + (1.33 + 0.0700i)18-s + (−0.169 + 0.379i)21-s + (0.113 − 0.255i)24-s + (0.453 − 0.891i)27-s + (−0.232 − 0.232i)28-s + ⋯
L(s)  = 1  + (0.946 + 0.946i)2-s + (0.933 − 0.358i)3-s + 0.790i·4-s + (1.22 + 0.544i)6-s + (−0.294 + 0.294i)7-s + (0.197 − 0.197i)8-s + (0.743 − 0.669i)9-s + (0.283 + 0.738i)12-s − 0.556·14-s + 1.16·16-s + (−0.147 − 0.147i)17-s + (1.33 + 0.0700i)18-s + (−0.169 + 0.379i)21-s + (0.113 − 0.255i)24-s + (0.453 − 0.891i)27-s + (−0.232 − 0.232i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.765 - 0.643i$
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: \((2,\ 3525,\ (\ :0),\ 0.765 - 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.960085487\)
\(L(\frac12)\) \(\approx\) \(2.960085487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.933 + 0.358i)T \)
5 \( 1 \)
47 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.946 - 0.946i)T + iT^{2} \)
7 \( 1 + (0.294 - 0.294i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.147 + 0.147i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
59 \( 1 + 0.415T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 0.813iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.61iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 - 1.90T + T^{2} \)
97 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
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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

−8.515701742733381909635979862413, −8.010738702621731594367310809270, −7.09970353139086932261555641065, −6.72160435065524883585104580786, −5.91503930172805132298270803040, −5.09312857884985086922940151298, −4.25025181073320961447714864638, −3.49113697146668056782372888886, −2.66638898452325577759332482596, −1.43849918948851363304652964546, 1.55550044316573569750855161159, 2.41813142918232470040513706689, 3.25491725180388055659185560720, 3.81699822775254464219827509503, 4.56930262652872993914885248583, 5.25590207856313516481596552850, 6.31071446139193666516549453312, 7.34293834595845539968468489751, 7.936509488674404276722766356005, 8.806481064602807023161208104897

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