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Properties

Label 2-3525-705.287-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.914 + 0.403i$
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  + (−0.123 − 0.0586i)2-s + (0.548 − 0.836i)3-s + (−0.619 − 0.761i)4-s + (−0.116 + 0.0709i)6-s + (0.0640 + 0.262i)8-s + (−0.398 − 0.917i)9-s + (−0.976 + 0.100i)12-s + (−0.192 + 0.924i)16-s + (1.09 + 0.348i)17-s + (−0.00465 + 0.136i)18-s + (−0.0861 − 1.25i)19-s + (−0.809 − 1.70i)23-s + (0.254 + 0.0905i)24-s + (−0.985 − 0.169i)27-s + (−1.73 − 0.361i)31-s + (0.241 − 0.318i)32-s + ⋯
L(s)  = 1  + (−0.123 − 0.0586i)2-s + (0.548 − 0.836i)3-s + (−0.619 − 0.761i)4-s + (−0.116 + 0.0709i)6-s + (0.0640 + 0.262i)8-s + (−0.398 − 0.917i)9-s + (−0.976 + 0.100i)12-s + (−0.192 + 0.924i)16-s + (1.09 + 0.348i)17-s + (−0.00465 + 0.136i)18-s + (−0.0861 − 1.25i)19-s + (−0.809 − 1.70i)23-s + (0.254 + 0.0905i)24-s + (−0.985 − 0.169i)27-s + (−1.73 − 0.361i)31-s + (0.241 − 0.318i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.914 + 0.403i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.914 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9900202527\)
\(L(\frac12)\) \(\approx\) \(0.9900202527\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.548 + 0.836i)T \)
5 \( 1 \)
47 \( 1 + (-0.871 + 0.490i)T \)
good2 \( 1 + (0.123 + 0.0586i)T + (0.631 + 0.775i)T^{2} \)
7 \( 1 + (0.942 - 0.334i)T^{2} \)
11 \( 1 + (-0.576 - 0.816i)T^{2} \)
13 \( 1 + (0.997 - 0.0682i)T^{2} \)
17 \( 1 + (-1.09 - 0.348i)T + (0.816 + 0.576i)T^{2} \)
19 \( 1 + (0.0861 + 1.25i)T + (-0.990 + 0.136i)T^{2} \)
23 \( 1 + (0.809 + 1.70i)T + (-0.631 + 0.775i)T^{2} \)
29 \( 1 + (0.0682 - 0.997i)T^{2} \)
31 \( 1 + (1.73 + 0.361i)T + (0.917 + 0.398i)T^{2} \)
37 \( 1 + (0.519 + 0.854i)T^{2} \)
41 \( 1 + (0.460 - 0.887i)T^{2} \)
43 \( 1 + (0.979 + 0.203i)T^{2} \)
53 \( 1 + (-1.78 - 0.434i)T + (0.887 + 0.460i)T^{2} \)
59 \( 1 + (0.203 + 0.979i)T^{2} \)
61 \( 1 + (1.31 + 0.368i)T + (0.854 + 0.519i)T^{2} \)
67 \( 1 + (0.942 + 0.334i)T^{2} \)
71 \( 1 + (0.775 + 0.631i)T^{2} \)
73 \( 1 + (-0.730 + 0.682i)T^{2} \)
79 \( 1 + (0.125 + 0.911i)T + (-0.962 + 0.269i)T^{2} \)
83 \( 1 + (1.62 - 0.516i)T + (0.816 - 0.576i)T^{2} \)
89 \( 1 + (-0.990 - 0.136i)T^{2} \)
97 \( 1 + (-0.398 - 0.917i)T^{2} \)
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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

−8.589182409362788009152760153406, −7.74083472067479894270815927167, −7.01901374878750359535935923749, −6.15565513171769956351513469455, −5.59207354373095964177771488922, −4.57705663898222568198987648420, −3.72332440547704697606116212615, −2.60934631980142703346571613942, −1.72412794387088999099103579945, −0.57046848586432002345905574556, 1.74168325631946456064063044316, 3.07079788090367403816811397740, 3.64894059114087714845726487846, 4.21728451926609937180268343372, 5.32737456735474406315458713052, 5.73809624353847911696028612938, 7.31431245304601299300645890433, 7.66834502581117256579698492478, 8.396396489562300403079062959748, 9.050565280272482502997282920485

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