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Properties

Label 2-3525-705.422-c0-0-16
Degree $2$
Conductor $3525$
Sign $0.848 - 0.529i$
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  + (−1.38 + 1.38i)2-s + (0.629 − 0.777i)3-s − 2.82i·4-s + (0.204 + 1.94i)6-s + (1.05 + 1.05i)7-s + (2.52 + 2.52i)8-s + (−0.207 − 0.978i)9-s + (−2.19 − 1.77i)12-s − 2.90·14-s − 4.16·16-s + (1.29 − 1.29i)17-s + (1.64 + 1.06i)18-s + (1.47 − 0.155i)21-s + (3.55 − 0.373i)24-s + (−0.891 − 0.453i)27-s + (2.97 − 2.97i)28-s + ⋯
L(s)  = 1  + (−1.38 + 1.38i)2-s + (0.629 − 0.777i)3-s − 2.82i·4-s + (0.204 + 1.94i)6-s + (1.05 + 1.05i)7-s + (2.52 + 2.52i)8-s + (−0.207 − 0.978i)9-s + (−2.19 − 1.77i)12-s − 2.90·14-s − 4.16·16-s + (1.29 − 1.29i)17-s + (1.64 + 1.06i)18-s + (1.47 − 0.155i)21-s + (3.55 − 0.373i)24-s + (−0.891 − 0.453i)27-s + (2.97 − 2.97i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9212627585\)
\(L(\frac12)\) \(\approx\) \(0.9212627585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.629 + 0.777i)T \)
5 \( 1 \)
47 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
7 \( 1 + (-1.05 - 1.05i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.29 + 1.29i)T - iT^{2} \)
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 \( 1 - 1.48T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.98iT - T^{2} \)
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

−8.631097366941371647091834995557, −7.911965438124791371409964353782, −7.69419579536020626324454334152, −6.85494377948411061220980521192, −6.06008376641393513496004941189, −5.43162572630327989215614082081, −4.70513805009049798983503639138, −2.86672139641619260053242342401, −1.88478541975768918415939562959, −1.01204117989195723452752370805, 1.16338293486493229686563358410, 1.97965720869226114456579282016, 2.99833870625151800619208709422, 3.94776017856347024375871875254, 4.16524637470549012478201014581, 5.40181349211346620072304518058, 7.06722348600112273362297680774, 7.68792717219304512066195807652, 8.342229366798108196862046518481, 8.567430551751128523432274806199

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