↑

Properties

Label 2-3525-141.2-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.995 - 0.0913i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0911 + 0.663i)2-s + (−0.519 − 0.854i)3-s + (0.530 + 0.148i)4-s + (0.614 − 0.266i)6-s + (−0.413 + 0.953i)8-s + (−0.460 + 0.887i)9-s + (−0.148 − 0.530i)12-s + (−0.123 − 0.0750i)16-s + (0.0997 − 0.0931i)17-s + (−0.547 − 0.386i)18-s + (0.644 − 1.81i)19-s + (−0.269 − 1.96i)23-s + (1.02 − 0.141i)24-s + (0.997 − 0.0682i)27-s + (1.16 + 0.709i)31-s + (−0.594 + 0.730i)32-s + ⋯
L(s)  = 1  + (−0.0911 + 0.663i)2-s + (−0.519 − 0.854i)3-s + (0.530 + 0.148i)4-s + (0.614 − 0.266i)6-s + (−0.413 + 0.953i)8-s + (−0.460 + 0.887i)9-s + (−0.148 − 0.530i)12-s + (−0.123 − 0.0750i)16-s + (0.0997 − 0.0931i)17-s + (−0.547 − 0.386i)18-s + (0.644 − 1.81i)19-s + (−0.269 − 1.96i)23-s + (1.02 − 0.141i)24-s + (0.997 − 0.0682i)27-s + (1.16 + 0.709i)31-s + (−0.594 + 0.730i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.995 - 0.0913i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1976, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.995 - 0.0913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196309294\)
\(L(\frac12)\) \(\approx\) \(1.196309294\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.519 + 0.854i)T \)
5 \( 1 \)
47 \( 1 + (-0.979 - 0.203i)T \)
good2 \( 1 + (0.0911 - 0.663i)T + (-0.962 - 0.269i)T^{2} \)
7 \( 1 + (-0.990 - 0.136i)T^{2} \)
11 \( 1 + (0.0682 + 0.997i)T^{2} \)
13 \( 1 + (-0.334 + 0.942i)T^{2} \)
17 \( 1 + (-0.0997 + 0.0931i)T + (0.0682 - 0.997i)T^{2} \)
19 \( 1 + (-0.644 + 1.81i)T + (-0.775 - 0.631i)T^{2} \)
23 \( 1 + (0.269 + 1.96i)T + (-0.962 + 0.269i)T^{2} \)
29 \( 1 + (0.334 + 0.942i)T^{2} \)
31 \( 1 + (-1.16 - 0.709i)T + (0.460 + 0.887i)T^{2} \)
37 \( 1 + (-0.917 + 0.398i)T^{2} \)
41 \( 1 + (-0.682 + 0.730i)T^{2} \)
43 \( 1 + (0.854 + 0.519i)T^{2} \)
53 \( 1 + (-0.366 - 0.843i)T + (-0.682 + 0.730i)T^{2} \)
59 \( 1 + (-0.854 + 0.519i)T^{2} \)
61 \( 1 + (0.234 - 1.12i)T + (-0.917 - 0.398i)T^{2} \)
67 \( 1 + (-0.990 + 0.136i)T^{2} \)
71 \( 1 + (-0.962 + 0.269i)T^{2} \)
73 \( 1 + (-0.576 + 0.816i)T^{2} \)
79 \( 1 + (-1.05 - 0.861i)T + (0.203 + 0.979i)T^{2} \)
83 \( 1 + (-1.34 - 1.25i)T + (0.0682 + 0.997i)T^{2} \)
89 \( 1 + (0.775 - 0.631i)T^{2} \)
97 \( 1 + (0.460 - 0.887i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.512799544055840361629293361495, −7.86853539186365705157951120475, −7.03830663735362681521347665029, −6.74916195287739593213983677224, −5.99005954924347810229344780807, −5.22163917377262574126088606255, −4.44872332602959205750328203641, −2.80660285596578912422612189492, −2.41679456237439879243577018334, −0.933467817325558041611637683001, 1.10825136144648223884522578575, 2.21902930601464369157671337477, 3.53724764068685637464089300417, 3.67622373037345200565620661018, 4.95335059042571660289434462437, 5.84090247022808253362495929319, 6.17822233002509206821224183686, 7.30356739641012440434884800452, 7.975891312711784800144225339633, 9.091194693521201830824930923980

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