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Properties

Label 2-3525-705.563-c0-0-5
Degree $2$
Conductor $3525$
Sign $0.646 - 0.762i$
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 + (0.891 + 0.453i)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 + (−0.734 + 1.44i)12-s + 1.90·14-s + (0.437 + 0.437i)17-s + (0.253 − 1.59i)18-s + (−1.11 + 0.363i)21-s + (0.951 − 0.309i)24-s + (0.156 + 0.987i)27-s + (−1.34 − 1.34i)28-s + ⋯
L(s)  = 1  + (−1.14 − 1.14i)2-s + (0.891 + 0.453i)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 + (−0.734 + 1.44i)12-s + 1.90·14-s + (0.437 + 0.437i)17-s + (0.253 − 1.59i)18-s + (−1.11 + 0.363i)21-s + (0.951 − 0.309i)24-s + (0.156 + 0.987i)27-s + (−1.34 − 1.34i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.646 - 0.762i$
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: \((2,\ 3525,\ (\ :0),\ 0.646 - 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7038812806\)
\(L(\frac12)\) \(\approx\) \(0.7038812806\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.891 - 0.453i)T \)
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

−9.055461225363784963947207893762, −8.411977360482367832301301010398, −7.88363473372293466016823706180, −6.90107401393402898643933306942, −5.87500659415461524054239603875, −4.84786439833948612220057906017, −3.63536878720010620694743144247, −3.14286833708921301787017889595, −2.37920531494847129494509772922, −1.48650650610145134318395422319, 0.54659724276299136973108284035, 1.75539062138956730014176721173, 3.13694949539534630183784631043, 3.81244308330635455807384296371, 5.09565852315987065636392034291, 6.16419519217095767232399288629, 6.74317500351549417355638532783, 7.29455531480306789669206184782, 7.83865627327475553884011074297, 8.527030813426590398098859094708

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