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Properties

Label 2-3525-705.323-c0-0-1
Degree $2$
Conductor $3525$
Sign $0.803 - 0.594i$
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  + (1.58 + 0.626i)2-s + (−0.657 − 0.753i)3-s + (1.40 + 1.31i)4-s + (−0.572 − 1.61i)6-s + (0.675 + 1.41i)8-s + (−0.136 + 0.990i)9-s + (0.0655 − 1.91i)12-s + (0.0523 + 0.764i)16-s + (0.0416 + 0.404i)17-s + (−0.837 + 1.48i)18-s + (1.24 + 0.759i)19-s + (0.292 + 0.741i)23-s + (0.626 − 1.44i)24-s + (0.836 − 0.548i)27-s + (1.25 − 0.0861i)31-s + (0.0792 − 0.249i)32-s + ⋯
L(s)  = 1  + (1.58 + 0.626i)2-s + (−0.657 − 0.753i)3-s + (1.40 + 1.31i)4-s + (−0.572 − 1.61i)6-s + (0.675 + 1.41i)8-s + (−0.136 + 0.990i)9-s + (0.0655 − 1.91i)12-s + (0.0523 + 0.764i)16-s + (0.0416 + 0.404i)17-s + (−0.837 + 1.48i)18-s + (1.24 + 0.759i)19-s + (0.292 + 0.741i)23-s + (0.626 − 1.44i)24-s + (0.836 − 0.548i)27-s + (1.25 − 0.0861i)31-s + (0.0792 − 0.249i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.803 - 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.634113635\)
\(L(\frac12)\) \(\approx\) \(2.634113635\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.657 + 0.753i)T \)
5 \( 1 \)
47 \( 1 + (-0.169 + 0.985i)T \)
good2 \( 1 + (-1.58 - 0.626i)T + (0.730 + 0.682i)T^{2} \)
7 \( 1 + (0.398 + 0.917i)T^{2} \)
11 \( 1 + (0.203 - 0.979i)T^{2} \)
13 \( 1 + (0.519 - 0.854i)T^{2} \)
17 \( 1 + (-0.0416 - 0.404i)T + (-0.979 + 0.203i)T^{2} \)
19 \( 1 + (-1.24 - 0.759i)T + (0.460 + 0.887i)T^{2} \)
23 \( 1 + (-0.292 - 0.741i)T + (-0.730 + 0.682i)T^{2} \)
29 \( 1 + (-0.854 + 0.519i)T^{2} \)
31 \( 1 + (-1.25 + 0.0861i)T + (0.990 - 0.136i)T^{2} \)
37 \( 1 + (-0.942 + 0.334i)T^{2} \)
41 \( 1 + (-0.775 + 0.631i)T^{2} \)
43 \( 1 + (-0.997 + 0.0682i)T^{2} \)
53 \( 1 + (1.78 + 0.850i)T + (0.631 + 0.775i)T^{2} \)
59 \( 1 + (-0.0682 + 0.997i)T^{2} \)
61 \( 1 + (-1.11 - 1.57i)T + (-0.334 + 0.942i)T^{2} \)
67 \( 1 + (0.398 - 0.917i)T^{2} \)
71 \( 1 + (-0.682 - 0.730i)T^{2} \)
73 \( 1 + (-0.269 - 0.962i)T^{2} \)
79 \( 1 + (1.37 - 0.713i)T + (0.576 - 0.816i)T^{2} \)
83 \( 1 + (0.0684 - 0.666i)T + (-0.979 - 0.203i)T^{2} \)
89 \( 1 + (0.460 - 0.887i)T^{2} \)
97 \( 1 + (-0.136 + 0.990i)T^{2} \)
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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

−8.301853800089225103512618883214, −7.79008633667077711413927690308, −6.98136980249003982403693126989, −6.54094410693403150140812638874, −5.58461343302715124378553444852, −5.37435292317222625827016087982, −4.41509318488156254906151012181, −3.52537219199944321945882877309, −2.64262651373464919597813899788, −1.41844372494821471759837146750, 1.17339396492180828394280719282, 2.72277448482522966027139498688, 3.20519710538161494749470174164, 4.23173723319737756198370870449, 4.80229007500272783502833826866, 5.30426272367725533931762931359, 6.20525451025676235850072175121, 6.67528400786136375809219027262, 7.75341893586895603866692302003, 8.953215975326141856073201382564

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