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Properties

Label 2-3525-705.362-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.054777504\)
\(L(\frac12)\) \(\approx\) \(1.054777504\)
\(L(1)\) 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

−8.798154886771190470085529412128, −8.233121547826155402013279868123, −7.54236377723423217367946689200, −6.91882866599824394585458819003, −5.31552639354102522139392609440, −4.26553071208603420093157220144, −3.37091099304242291825622162579, −3.10593211946689206590662927629, −1.86895877451711244733411127120, −1.29169134641106067670979185475, 0.928923225037744778661145064634, 2.16022151722610807176954959437, 3.45873941533132729740455297238, 4.65298430394581533030231933491, 5.21220037444523104182788377173, 6.24940208004135761483634333195, 6.91394314243104998188224330004, 7.53782690704587696051418612745, 8.104216567519885443961491385911, 8.709927891539018186451749305367

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