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Properties

Label 2-3525-141.101-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.756 + 0.653i$
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  + (1.15 + 0.0787i)2-s + (−0.269 − 0.962i)3-s + (0.327 + 0.0449i)4-s + (−0.234 − 1.12i)6-s + (−0.756 − 0.157i)8-s + (−0.854 + 0.519i)9-s + (−0.0449 − 0.327i)12-s + (−1.17 − 0.329i)16-s + (0.543 − 1.25i)17-s + (−1.02 + 0.530i)18-s + (−1.14 − 1.61i)19-s + (0.136 − 0.00931i)23-s + (0.0527 + 0.770i)24-s + (0.730 + 0.682i)27-s + (−1.76 − 0.494i)31-s + (−0.599 − 0.212i)32-s + ⋯
L(s)  = 1  + (1.15 + 0.0787i)2-s + (−0.269 − 0.962i)3-s + (0.327 + 0.0449i)4-s + (−0.234 − 1.12i)6-s + (−0.756 − 0.157i)8-s + (−0.854 + 0.519i)9-s + (−0.0449 − 0.327i)12-s + (−1.17 − 0.329i)16-s + (0.543 − 1.25i)17-s + (−1.02 + 0.530i)18-s + (−1.14 − 1.61i)19-s + (0.136 − 0.00931i)23-s + (0.0527 + 0.770i)24-s + (0.730 + 0.682i)27-s + (−1.76 − 0.494i)31-s + (−0.599 − 0.212i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.756 + 0.653i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.756 + 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267383564\)
\(L(\frac12)\) \(\approx\) \(1.267383564\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.269 + 0.962i)T \)
5 \( 1 \)
47 \( 1 + (0.631 + 0.775i)T \)
good2 \( 1 + (-1.15 - 0.0787i)T + (0.990 + 0.136i)T^{2} \)
7 \( 1 + (-0.0682 + 0.997i)T^{2} \)
11 \( 1 + (-0.682 + 0.730i)T^{2} \)
13 \( 1 + (-0.576 - 0.816i)T^{2} \)
17 \( 1 + (-0.543 + 1.25i)T + (-0.682 - 0.730i)T^{2} \)
19 \( 1 + (1.14 + 1.61i)T + (-0.334 + 0.942i)T^{2} \)
23 \( 1 + (-0.136 + 0.00931i)T + (0.990 - 0.136i)T^{2} \)
29 \( 1 + (0.576 - 0.816i)T^{2} \)
31 \( 1 + (1.76 + 0.494i)T + (0.854 + 0.519i)T^{2} \)
37 \( 1 + (0.203 + 0.979i)T^{2} \)
41 \( 1 + (0.917 - 0.398i)T^{2} \)
43 \( 1 + (0.962 + 0.269i)T^{2} \)
53 \( 1 + (-1.67 + 0.347i)T + (0.917 - 0.398i)T^{2} \)
59 \( 1 + (-0.962 + 0.269i)T^{2} \)
61 \( 1 + (0.713 - 0.580i)T + (0.203 - 0.979i)T^{2} \)
67 \( 1 + (-0.0682 - 0.997i)T^{2} \)
71 \( 1 + (0.990 - 0.136i)T^{2} \)
73 \( 1 + (0.460 + 0.887i)T^{2} \)
79 \( 1 + (0.614 - 1.72i)T + (-0.775 - 0.631i)T^{2} \)
83 \( 1 + (-0.162 - 0.373i)T + (-0.682 + 0.730i)T^{2} \)
89 \( 1 + (0.334 + 0.942i)T^{2} \)
97 \( 1 + (0.854 - 0.519i)T^{2} \)
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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

−8.493663723776613672042813598570, −7.23187987573129897086587591485, −7.01172635683641973669078049330, −6.11054588712552064780249600452, −5.35950284247333791278636625589, −4.87645086510985568759477462448, −3.85602448223505082842812543511, −2.87307112598981555171915803332, −2.14952114980991995147659223217, −0.49191787550514149217924917757, 1.87756890639676359228947425576, 3.18633303540851939334440206960, 3.77026990139490227090078914279, 4.33779520772362133800434191897, 5.16631724360257568707358188213, 6.01844541494873152847945172350, 6.13874289378810287932598095392, 7.52677972949873783944542408770, 8.512486990483736366346645297629, 8.954393420201560258705419986841

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