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Properties

Label 2-3525-705.302-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.333 - 0.942i$
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  + (−0.869 − 0.757i)2-s + (−0.490 − 0.871i)3-s + (0.0449 + 0.327i)4-s + (−0.234 + 1.12i)6-s + (−0.423 + 0.645i)8-s + (−0.519 + 0.854i)9-s + (0.263 − 0.199i)12-s + (1.17 − 0.329i)16-s + (0.500 − 1.26i)17-s + (1.09 − 0.348i)18-s + (−0.157 + 0.222i)19-s + (−1.31 − 1.50i)23-s + (0.770 + 0.0527i)24-s + (0.999 + 0.0341i)27-s + (−0.214 − 0.767i)31-s + (−0.574 − 0.273i)32-s + ⋯
L(s)  = 1  + (−0.869 − 0.757i)2-s + (−0.490 − 0.871i)3-s + (0.0449 + 0.327i)4-s + (−0.234 + 1.12i)6-s + (−0.423 + 0.645i)8-s + (−0.519 + 0.854i)9-s + (0.263 − 0.199i)12-s + (1.17 − 0.329i)16-s + (0.500 − 1.26i)17-s + (1.09 − 0.348i)18-s + (−0.157 + 0.222i)19-s + (−1.31 − 1.50i)23-s + (0.770 + 0.0527i)24-s + (0.999 + 0.0341i)27-s + (−0.214 − 0.767i)31-s + (−0.574 − 0.273i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.333 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2232771914\)
\(L(\frac12)\) \(\approx\) \(0.2232771914\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.490 + 0.871i)T \)
5 \( 1 \)
47 \( 1 + (0.994 - 0.102i)T \)
good2 \( 1 + (0.869 + 0.757i)T + (0.136 + 0.990i)T^{2} \)
7 \( 1 + (0.997 - 0.0682i)T^{2} \)
11 \( 1 + (0.682 + 0.730i)T^{2} \)
13 \( 1 + (-0.816 - 0.576i)T^{2} \)
17 \( 1 + (-0.500 + 1.26i)T + (-0.730 - 0.682i)T^{2} \)
19 \( 1 + (0.157 - 0.222i)T + (-0.334 - 0.942i)T^{2} \)
23 \( 1 + (1.31 + 1.50i)T + (-0.136 + 0.990i)T^{2} \)
29 \( 1 + (0.576 + 0.816i)T^{2} \)
31 \( 1 + (0.214 + 0.767i)T + (-0.854 + 0.519i)T^{2} \)
37 \( 1 + (0.979 + 0.203i)T^{2} \)
41 \( 1 + (-0.917 - 0.398i)T^{2} \)
43 \( 1 + (0.269 + 0.962i)T^{2} \)
53 \( 1 + (1.42 - 0.937i)T + (0.398 - 0.917i)T^{2} \)
59 \( 1 + (0.962 + 0.269i)T^{2} \)
61 \( 1 + (-0.713 - 0.580i)T + (0.203 + 0.979i)T^{2} \)
67 \( 1 + (0.997 + 0.0682i)T^{2} \)
71 \( 1 + (0.990 + 0.136i)T^{2} \)
73 \( 1 + (0.887 + 0.460i)T^{2} \)
79 \( 1 + (1.72 - 0.614i)T + (0.775 - 0.631i)T^{2} \)
83 \( 1 + (-0.149 - 0.378i)T + (-0.730 + 0.682i)T^{2} \)
89 \( 1 + (-0.334 + 0.942i)T^{2} \)
97 \( 1 + (-0.519 + 0.854i)T^{2} \)
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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

−8.166056147239214428988206990527, −7.79851929505710663053428137819, −6.78953816932167797123890045797, −6.06584464226787290649172578895, −5.35466247248051283168126178678, −4.43707105529223128791075426339, −2.99537667285412130852280894730, −2.28925473666108062816722687081, −1.35772463452460671209703398165, −0.19567414072179363521271503210, 1.56068956208228097966558170318, 3.32032469962804337684415339030, 3.77283126746183638734765690538, 4.84471678044773833834836427515, 5.77611205936946085547676708771, 6.26181980854357610768212647113, 7.06950692290961775991349217290, 8.028408150622726435134173923510, 8.398337359315300225756100035115, 9.305497272553913434912716134882

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