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Properties

Label 2-3525-141.32-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.653 - 0.756i$
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  + (1.25 − 1.53i)2-s + (0.398 − 0.917i)3-s + (−0.595 − 2.86i)4-s + (−0.911 − 1.75i)6-s + (−3.38 − 1.75i)8-s + (−0.682 − 0.730i)9-s + (−2.86 − 0.595i)12-s + (−4.25 + 1.84i)16-s + (0.547 − 0.386i)17-s + (−1.97 + 0.135i)18-s + (0.403 + 0.0554i)19-s + (0.979 + 1.20i)23-s + (−2.95 + 2.40i)24-s + (−0.942 + 0.334i)27-s + (1.05 − 0.459i)31-s + (−1.44 + 5.16i)32-s + ⋯
L(s)  = 1  + (1.25 − 1.53i)2-s + (0.398 − 0.917i)3-s + (−0.595 − 2.86i)4-s + (−0.911 − 1.75i)6-s + (−3.38 − 1.75i)8-s + (−0.682 − 0.730i)9-s + (−2.86 − 0.595i)12-s + (−4.25 + 1.84i)16-s + (0.547 − 0.386i)17-s + (−1.97 + 0.135i)18-s + (0.403 + 0.0554i)19-s + (0.979 + 1.20i)23-s + (−2.95 + 2.40i)24-s + (−0.942 + 0.334i)27-s + (1.05 − 0.459i)31-s + (−1.44 + 5.16i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.653 - 0.756i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.653 - 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.474915386\)
\(L(\frac12)\) \(\approx\) \(2.474915386\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.398 + 0.917i)T \)
5 \( 1 \)
47 \( 1 + (0.519 + 0.854i)T \)
good2 \( 1 + (-1.25 + 1.53i)T + (-0.203 - 0.979i)T^{2} \)
7 \( 1 + (-0.775 - 0.631i)T^{2} \)
11 \( 1 + (0.334 + 0.942i)T^{2} \)
13 \( 1 + (-0.990 - 0.136i)T^{2} \)
17 \( 1 + (-0.547 + 0.386i)T + (0.334 - 0.942i)T^{2} \)
19 \( 1 + (-0.403 - 0.0554i)T + (0.962 + 0.269i)T^{2} \)
23 \( 1 + (-0.979 - 1.20i)T + (-0.203 + 0.979i)T^{2} \)
29 \( 1 + (0.990 - 0.136i)T^{2} \)
31 \( 1 + (-1.05 + 0.459i)T + (0.682 - 0.730i)T^{2} \)
37 \( 1 + (0.460 + 0.887i)T^{2} \)
41 \( 1 + (0.576 - 0.816i)T^{2} \)
43 \( 1 + (-0.917 + 0.398i)T^{2} \)
53 \( 1 + (1.21 - 0.628i)T + (0.576 - 0.816i)T^{2} \)
59 \( 1 + (0.917 + 0.398i)T^{2} \)
61 \( 1 + (0.116 - 0.0709i)T + (0.460 - 0.887i)T^{2} \)
67 \( 1 + (-0.775 + 0.631i)T^{2} \)
71 \( 1 + (-0.203 + 0.979i)T^{2} \)
73 \( 1 + (-0.0682 - 0.997i)T^{2} \)
79 \( 1 + (-1.11 - 0.311i)T + (0.854 + 0.519i)T^{2} \)
83 \( 1 + (0.751 + 0.530i)T + (0.334 + 0.942i)T^{2} \)
89 \( 1 + (-0.962 + 0.269i)T^{2} \)
97 \( 1 + (0.682 + 0.730i)T^{2} \)
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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

−8.433426632352227131120215686695, −7.38278655931310568368873300871, −6.56451771744759474717001585594, −5.75933968113928434942652279981, −5.16957789813914569173224672009, −4.18641260119732288312039129318, −3.24217696208277564264913959749, −2.81269305678493562082755078134, −1.75545956810703572702982044863, −0.952846316674339823781155089700, 2.63353637751582315534780447259, 3.27657350567072033161872527489, 4.07683316117363246788054354800, 4.81709081664465919684575599177, 5.27868171087978381172049244203, 6.18221648769403152066780812861, 6.80565430767156302561278510802, 7.74595529142508797135042633847, 8.261010482494543379917985221229, 8.886231512697818482490970159804

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