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Properties

Label 2-3525-705.563-c0-0-0
Degree $2$
Conductor $3525$
Sign $-0.525 - 0.850i$
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  + (−1.41 − 1.41i)2-s + (−0.707 + 0.707i)3-s + 3.00i·4-s + 2.00·6-s + (2.82 − 2.82i)8-s − 1.00i·9-s + (−2.12 − 2.12i)12-s − 5.00·16-s + (−1.41 − 1.41i)17-s + (−1.41 + 1.41i)18-s + 4.00i·24-s + (0.707 + 0.707i)27-s + (4.24 + 4.24i)32-s + 4.00i·34-s + 3.00·36-s + ⋯
L(s)  = 1  + (−1.41 − 1.41i)2-s + (−0.707 + 0.707i)3-s + 3.00i·4-s + 2.00·6-s + (2.82 − 2.82i)8-s − 1.00i·9-s + (−2.12 − 2.12i)12-s − 5.00·16-s + (−1.41 − 1.41i)17-s + (−1.41 + 1.41i)18-s + 4.00i·24-s + (0.707 + 0.707i)27-s + (4.24 + 4.24i)32-s + 4.00i·34-s + 3.00·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.525 - 0.850i$
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: \((2,\ 3525,\ (\ :0),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02246128886\)
\(L(\frac12)\) \(\approx\) \(0.02246128886\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
47 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
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 \( 1 + T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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

−9.290007144174539222400448215427, −8.734456501811200227665466984482, −7.70811949796054439109430297998, −7.07051961613847778697651889992, −6.20252159711085358952016837714, −4.70614792369636978201360052190, −4.32406495742897439624450201081, −3.21752879744645662019537323083, −2.54551650536818503774380011619, −1.26686764090629945135356212566, 0.02550621541040792553574031445, 1.45111142254822729436899561053, 2.17942958723463595314147947952, 4.34031028491208446841728372341, 5.15330539820030357651191261394, 5.92279730469585319432840119147, 6.48418160989748299374815996945, 6.97332102915814651350596777234, 7.76023903959696619050023786516, 8.386745392095143630414674908003

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