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Properties

Label 2-3525-705.317-c0-0-0
Degree $2$
Conductor $3525$
Sign $-0.909 - 0.415i$
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  + (−0.760 + 1.35i)2-s + (−0.302 + 0.953i)3-s + (−0.731 − 1.20i)4-s + (−1.05 − 1.13i)6-s + (0.630 − 0.0215i)8-s + (−0.816 − 0.576i)9-s + (1.36 − 0.333i)12-s + (0.196 − 0.379i)16-s + (1.49 + 1.30i)17-s + (1.40 − 0.666i)18-s + (0.806 − 0.655i)19-s + (0.470 − 0.264i)23-s + (−0.170 + 0.607i)24-s + (0.796 − 0.604i)27-s + (1.77 + 0.917i)31-s + (0.709 + 1.08i)32-s + ⋯
L(s)  = 1  + (−0.760 + 1.35i)2-s + (−0.302 + 0.953i)3-s + (−0.731 − 1.20i)4-s + (−1.05 − 1.13i)6-s + (0.630 − 0.0215i)8-s + (−0.816 − 0.576i)9-s + (1.36 − 0.333i)12-s + (0.196 − 0.379i)16-s + (1.49 + 1.30i)17-s + (1.40 − 0.666i)18-s + (0.806 − 0.655i)19-s + (0.470 − 0.264i)23-s + (−0.170 + 0.607i)24-s + (0.796 − 0.604i)27-s + (1.77 + 0.917i)31-s + (0.709 + 1.08i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.909 - 0.415i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7497591709\)
\(L(\frac12)\) \(\approx\) \(0.7497591709\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.302 - 0.953i)T \)
5 \( 1 \)
47 \( 1 + (-0.366 + 0.930i)T \)
good2 \( 1 + (0.760 - 1.35i)T + (-0.519 - 0.854i)T^{2} \)
7 \( 1 + (0.269 + 0.962i)T^{2} \)
11 \( 1 + (-0.990 - 0.136i)T^{2} \)
13 \( 1 + (-0.631 - 0.775i)T^{2} \)
17 \( 1 + (-1.49 - 1.30i)T + (0.136 + 0.990i)T^{2} \)
19 \( 1 + (-0.806 + 0.655i)T + (0.203 - 0.979i)T^{2} \)
23 \( 1 + (-0.470 + 0.264i)T + (0.519 - 0.854i)T^{2} \)
29 \( 1 + (0.775 + 0.631i)T^{2} \)
31 \( 1 + (-1.77 - 0.917i)T + (0.576 + 0.816i)T^{2} \)
37 \( 1 + (-0.730 + 0.682i)T^{2} \)
41 \( 1 + (-0.0682 + 0.997i)T^{2} \)
43 \( 1 + (0.887 + 0.460i)T^{2} \)
53 \( 1 + (-0.0393 + 1.15i)T + (-0.997 - 0.0682i)T^{2} \)
59 \( 1 + (0.460 + 0.887i)T^{2} \)
61 \( 1 + (0.614 - 0.266i)T + (0.682 - 0.730i)T^{2} \)
67 \( 1 + (0.269 - 0.962i)T^{2} \)
71 \( 1 + (-0.854 - 0.519i)T^{2} \)
73 \( 1 + (-0.942 - 0.334i)T^{2} \)
79 \( 1 + (0.133 + 0.0277i)T + (0.917 + 0.398i)T^{2} \)
83 \( 1 + (1.02 - 0.897i)T + (0.136 - 0.990i)T^{2} \)
89 \( 1 + (0.203 + 0.979i)T^{2} \)
97 \( 1 + (-0.816 - 0.576i)T^{2} \)
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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

−8.814969222448153497955204885963, −8.424469062760214984527694923709, −7.68845414814176052087530173942, −6.77969911108947803432500687861, −6.19621238953506696989769530191, −5.37950828674367019517373670554, −4.93476298851741638382574777507, −3.75562614890927256233756819935, −2.92635314609810490340467317674, −1.00120296574282918765700566182, 0.837947988832501435980131108111, 1.51483570800207957167076082284, 2.79920546883134351237772845871, 3.07363312821281487115247081867, 4.43892596613621704775807410004, 5.52734602593780261588849136579, 6.14974472059056167899064428993, 7.31166253908836831308416222524, 7.78283548446209315000544669744, 8.457557692204563619359505818901

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