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Properties

Label 2-3525-705.422-c0-0-14
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: $\chi_{3525} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.646 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.994146304\)
\(L(\frac12)\) \(\approx\) \(1.994146304\)
\(L(1)\) 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

−8.840560457636217420773971301291, −7.994045245457298356653322214981, −6.82786611353287901644162565281, −5.87429826073502142270125284948, −5.50289041865698708816646304301, −4.61109482687593992136798640666, −4.24773921251716701762991926081, −3.18537833496421731620429209274, −2.23697246251203749123268717835, −1.27530065763881183560254966616, 1.11550402676002248517767201925, 2.51863708552352497970132548478, 4.06608416210623758947861418484, 4.33527532452211928671698805646, 5.30315569301860987681235758710, 5.69097070680443832627514337718, 6.69459170445996222052084749473, 7.08489424180831903691396562391, 7.75951567093428211191023820581, 8.315134255267933022568794722386

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