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Properties

Label 2-3525-705.563-c0-0-20
Degree $2$
Conductor $3525$
Sign $-0.997 + 0.0706i$
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  + (−0.707 − 0.707i)2-s + (−0.965 − 0.258i)3-s + (0.500 + 0.866i)6-s + (1.22 − 1.22i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s − 1.73·14-s + 1.00·16-s + (−0.707 − 0.707i)17-s + (−0.258 − 0.965i)18-s + (−1.49 + 0.866i)21-s + (0.866 − 0.5i)24-s + (−0.707 − 0.707i)27-s + 1.00i·34-s + (1.67 + 0.448i)42-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.965 − 0.258i)3-s + (0.500 + 0.866i)6-s + (1.22 − 1.22i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s − 1.73·14-s + 1.00·16-s + (−0.707 − 0.707i)17-s + (−0.258 − 0.965i)18-s + (−1.49 + 0.866i)21-s + (0.866 − 0.5i)24-s + (−0.707 − 0.707i)27-s + 1.00i·34-s + (1.67 + 0.448i)42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.997 + 0.0706i$
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: \((2,\ 3525,\ (\ :0),\ -0.997 + 0.0706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5688138034\)
\(L(\frac12)\) \(\approx\) \(0.5688138034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
47 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.409527735808738301341594477190, −7.67457326184763803167278552069, −7.01508270515550724665382724554, −6.21292831485641222882476799563, −5.15836582551462524161147868387, −4.76820052913378213234026194602, −3.74252362437174128166398104950, −2.25806995042473473997359765252, −1.47170732745614127750193532824, −0.52990121037317696304302490013, 1.38117590986210071406816170902, 2.59494605629484204668092222342, 3.94987393805511547949300593783, 4.71693421016366261823810357433, 5.62966852900405366802274665997, 6.06555696298735541017623365816, 6.94071762267994260356309267057, 7.64682232546568317055654885819, 8.506531953241660609151842512921, 8.823336694435439102303457430451

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