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Properties

Label 2-3525-705.137-c0-0-1
Degree $2$
Conductor $3525$
Sign $0.778 + 0.627i$
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  + (−1.35 + 0.760i)2-s + (0.953 − 0.302i)3-s + (0.731 − 1.20i)4-s + (−1.05 + 1.13i)6-s + (−0.0215 + 0.630i)8-s + (0.816 − 0.576i)9-s + (0.333 − 1.36i)12-s + (0.196 + 0.379i)16-s + (−1.30 − 1.49i)17-s + (−0.666 + 1.40i)18-s + (−0.806 − 0.655i)19-s + (−0.264 + 0.470i)23-s + (0.170 + 0.607i)24-s + (0.604 − 0.796i)27-s + (1.77 − 0.917i)31-s + (−1.08 − 0.709i)32-s + ⋯
L(s)  = 1  + (−1.35 + 0.760i)2-s + (0.953 − 0.302i)3-s + (0.731 − 1.20i)4-s + (−1.05 + 1.13i)6-s + (−0.0215 + 0.630i)8-s + (0.816 − 0.576i)9-s + (0.333 − 1.36i)12-s + (0.196 + 0.379i)16-s + (−1.30 − 1.49i)17-s + (−0.666 + 1.40i)18-s + (−0.806 − 0.655i)19-s + (−0.264 + 0.470i)23-s + (0.170 + 0.607i)24-s + (0.604 − 0.796i)27-s + (1.77 − 0.917i)31-s + (−1.08 − 0.709i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2957, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.778 + 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7754277166\)
\(L(\frac12)\) \(\approx\) \(0.7754277166\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.953 + 0.302i)T \)
5 \( 1 \)
47 \( 1 + (-0.930 + 0.366i)T \)
good2 \( 1 + (1.35 - 0.760i)T + (0.519 - 0.854i)T^{2} \)
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 \( 1 + (1.30 + 1.49i)T + (-0.136 + 0.990i)T^{2} \)
19 \( 1 + (0.806 + 0.655i)T + (0.203 + 0.979i)T^{2} \)
23 \( 1 + (0.264 - 0.470i)T + (-0.519 - 0.854i)T^{2} \)
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 \( 1 + (1.15 - 0.0393i)T + (0.997 - 0.0682i)T^{2} \)
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 \( 1 + (-0.897 + 1.02i)T + (-0.136 - 0.990i)T^{2} \)
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

−8.738785027868642760331211354720, −7.954896152194681155816024153341, −7.41743281677374518279257020659, −6.69229158924348988119669590267, −6.25665350560053665176774397517, −4.87840809699714774911028942028, −4.02903879390656318898872696710, −2.77708951910562436090898376329, −1.99516074997211080972082153642, −0.64203265038007223022020975835, 1.41974039366425969535799070342, 2.20038844113682772704554248602, 2.95159805414169929782989902674, 3.99863725327907192032842706353, 4.66392130919808536961549630106, 6.08961189882221721853668088188, 6.87614916222352215104035779921, 7.920162861466202707237183995201, 8.312605698885753165388676154951, 8.814781174104111795807884355324

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