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Properties

Label 2-3525-141.56-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.510 - 0.859i$
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  + (−1.62 − 0.843i)2-s + (−0.942 + 0.334i)3-s + (1.36 + 1.93i)4-s + (1.81 + 0.249i)6-s + (−0.340 − 2.48i)8-s + (0.775 − 0.631i)9-s + (−1.93 − 1.36i)12-s + (−0.746 + 2.09i)16-s + (0.461 + 1.64i)17-s + (−1.79 + 0.373i)18-s + (−1.05 − 0.459i)19-s + (0.816 − 0.423i)23-s + (1.15 + 2.22i)24-s + (−0.519 + 0.854i)27-s + (−0.644 + 1.81i)31-s + (1.15 − 1.08i)32-s + ⋯
L(s)  = 1  + (−1.62 − 0.843i)2-s + (−0.942 + 0.334i)3-s + (1.36 + 1.93i)4-s + (1.81 + 0.249i)6-s + (−0.340 − 2.48i)8-s + (0.775 − 0.631i)9-s + (−1.93 − 1.36i)12-s + (−0.746 + 2.09i)16-s + (0.461 + 1.64i)17-s + (−1.79 + 0.373i)18-s + (−1.05 − 0.459i)19-s + (0.816 − 0.423i)23-s + (1.15 + 2.22i)24-s + (−0.519 + 0.854i)27-s + (−0.644 + 1.81i)31-s + (1.15 − 1.08i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.510 - 0.859i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2876, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.510 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2964215004\)
\(L(\frac12)\) \(\approx\) \(0.2964215004\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.942 - 0.334i)T \)
5 \( 1 \)
47 \( 1 + (-0.997 + 0.0682i)T \)
good2 \( 1 + (1.62 + 0.843i)T + (0.576 + 0.816i)T^{2} \)
7 \( 1 + (0.460 - 0.887i)T^{2} \)
11 \( 1 + (-0.854 - 0.519i)T^{2} \)
13 \( 1 + (-0.917 - 0.398i)T^{2} \)
17 \( 1 + (-0.461 - 1.64i)T + (-0.854 + 0.519i)T^{2} \)
19 \( 1 + (1.05 + 0.459i)T + (0.682 + 0.730i)T^{2} \)
23 \( 1 + (-0.816 + 0.423i)T + (0.576 - 0.816i)T^{2} \)
29 \( 1 + (0.917 - 0.398i)T^{2} \)
31 \( 1 + (0.644 - 1.81i)T + (-0.775 - 0.631i)T^{2} \)
37 \( 1 + (-0.990 - 0.136i)T^{2} \)
41 \( 1 + (-0.962 - 0.269i)T^{2} \)
43 \( 1 + (-0.334 + 0.942i)T^{2} \)
53 \( 1 + (-0.211 + 1.53i)T + (-0.962 - 0.269i)T^{2} \)
59 \( 1 + (0.334 + 0.942i)T^{2} \)
61 \( 1 + (0.0277 + 0.405i)T + (-0.990 + 0.136i)T^{2} \)
67 \( 1 + (0.460 + 0.887i)T^{2} \)
71 \( 1 + (0.576 - 0.816i)T^{2} \)
73 \( 1 + (0.203 + 0.979i)T^{2} \)
79 \( 1 + (1.31 + 1.40i)T + (-0.0682 + 0.997i)T^{2} \)
83 \( 1 + (0.534 - 1.90i)T + (-0.854 - 0.519i)T^{2} \)
89 \( 1 + (-0.682 + 0.730i)T^{2} \)
97 \( 1 + (-0.775 + 0.631i)T^{2} \)
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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

−8.810079510979434656275188131426, −8.560909770674911589722334102145, −7.52347708174650168725506481048, −6.80813940669473427676185856161, −6.16721899527771552096138119220, −5.06025263748096281754405538576, −4.01066520408116445720386536111, −3.21494592687388261140318033637, −1.98588493426346270147763036347, −1.08861157921091263229000581260, 0.40883052460541751673243566251, 1.50452041843492128421148664494, 2.56486398952602746045691568891, 4.28164836011523494013083487592, 5.35963986767571363462716811736, 5.84562697369526993424083433773, 6.62571542432695459535569783732, 7.36912747207386419721884208397, 7.61998488139521418725996668792, 8.607218517841331332048423924508

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