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Properties

Label 2-3525-705.362-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.432 + 0.901i$
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  + (1.05 + 1.61i)2-s + (−0.999 + 0.0341i)3-s + (−1.07 + 2.48i)4-s + (−1.11 − 1.57i)6-s + (−3.24 + 0.559i)8-s + (0.997 − 0.0682i)9-s + (0.993 − 2.51i)12-s + (−2.47 − 2.65i)16-s + (−1.40 − 0.666i)17-s + (1.16 + 1.53i)18-s + (−0.767 + 0.214i)19-s + (−1.63 − 1.07i)23-s + (3.22 − 0.669i)24-s + (−0.994 + 0.102i)27-s + (−1.37 + 1.28i)31-s + (0.875 − 3.59i)32-s + ⋯
L(s)  = 1  + (1.05 + 1.61i)2-s + (−0.999 + 0.0341i)3-s + (−1.07 + 2.48i)4-s + (−1.11 − 1.57i)6-s + (−3.24 + 0.559i)8-s + (0.997 − 0.0682i)9-s + (0.993 − 2.51i)12-s + (−2.47 − 2.65i)16-s + (−1.40 − 0.666i)17-s + (1.16 + 1.53i)18-s + (−0.767 + 0.214i)19-s + (−1.63 − 1.07i)23-s + (3.22 − 0.669i)24-s + (−0.994 + 0.102i)27-s + (−1.37 + 1.28i)31-s + (0.875 − 3.59i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.432 + 0.901i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.432 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3442344594\)
\(L(\frac12)\) \(\approx\) \(0.3442344594\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.999 - 0.0341i)T \)
5 \( 1 \)
47 \( 1 + (-0.953 - 0.302i)T \)
good2 \( 1 + (-1.05 - 1.61i)T + (-0.398 + 0.917i)T^{2} \)
7 \( 1 + (0.979 + 0.203i)T^{2} \)
11 \( 1 + (-0.775 - 0.631i)T^{2} \)
13 \( 1 + (0.269 + 0.962i)T^{2} \)
17 \( 1 + (1.40 + 0.666i)T + (0.631 + 0.775i)T^{2} \)
19 \( 1 + (0.767 - 0.214i)T + (0.854 - 0.519i)T^{2} \)
23 \( 1 + (1.63 + 1.07i)T + (0.398 + 0.917i)T^{2} \)
29 \( 1 + (-0.962 - 0.269i)T^{2} \)
31 \( 1 + (1.37 - 1.28i)T + (0.0682 - 0.997i)T^{2} \)
37 \( 1 + (0.816 - 0.576i)T^{2} \)
41 \( 1 + (-0.334 + 0.942i)T^{2} \)
43 \( 1 + (-0.730 + 0.682i)T^{2} \)
53 \( 1 + (-0.0231 + 0.134i)T + (-0.942 - 0.334i)T^{2} \)
59 \( 1 + (0.682 - 0.730i)T^{2} \)
61 \( 1 + (-0.911 - 1.75i)T + (-0.576 + 0.816i)T^{2} \)
67 \( 1 + (0.979 - 0.203i)T^{2} \)
71 \( 1 + (0.917 - 0.398i)T^{2} \)
73 \( 1 + (0.136 - 0.990i)T^{2} \)
79 \( 1 + (0.347 + 0.572i)T + (-0.460 + 0.887i)T^{2} \)
83 \( 1 + (1.04 - 0.495i)T + (0.631 - 0.775i)T^{2} \)
89 \( 1 + (0.854 + 0.519i)T^{2} \)
97 \( 1 + (0.997 - 0.0682i)T^{2} \)
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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

−8.928696734306184576578540481801, −8.446468177843768131585230799417, −7.43432840703545223558670867945, −6.93350919319328879148573345335, −6.27799293428066392032341960861, −5.76742187322937162159858937428, −4.89513934071005935355260154537, −4.37929628469240355685074545400, −3.69260553126055024337494400334, −2.29819926376367159074863972963, 0.14886612823356791346228153996, 1.72688418612930433620367830952, 2.21191585126451894952869254514, 3.67373224917017571589518045966, 4.15096353904116377644445941069, 4.84487587775675176685065714204, 5.80451098658665480642283398103, 6.09545514470808911755458819785, 7.08756933085214611778655966081, 8.342069473078183780193242666181

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