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Properties

Label 2-3525-705.23-c0-0-1
Degree $2$
Conductor $3525$
Sign $0.788 + 0.615i$
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  + (0.533 + 0.404i)2-s + (0.971 − 0.236i)3-s + (−0.148 − 0.530i)4-s + (0.614 + 0.266i)6-s + (0.381 − 0.966i)8-s + (0.887 − 0.460i)9-s + (−0.270 − 0.480i)12-s + (0.123 − 0.0750i)16-s + (−0.136 + 0.00465i)17-s + (0.660 + 0.113i)18-s + (−0.180 − 0.508i)19-s + (−0.164 − 0.216i)23-s + (0.141 − 1.02i)24-s + (0.753 − 0.657i)27-s + (0.759 + 1.24i)31-s + (−0.937 − 0.0963i)32-s + ⋯
L(s)  = 1  + (0.533 + 0.404i)2-s + (0.971 − 0.236i)3-s + (−0.148 − 0.530i)4-s + (0.614 + 0.266i)6-s + (0.381 − 0.966i)8-s + (0.887 − 0.460i)9-s + (−0.270 − 0.480i)12-s + (0.123 − 0.0750i)16-s + (−0.136 + 0.00465i)17-s + (0.660 + 0.113i)18-s + (−0.180 − 0.508i)19-s + (−0.164 − 0.216i)23-s + (0.141 − 1.02i)24-s + (0.753 − 0.657i)27-s + (0.759 + 1.24i)31-s + (−0.937 − 0.0963i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.317316918\)
\(L(\frac12)\) \(\approx\) \(2.317316918\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.971 + 0.236i)T \)
5 \( 1 \)
47 \( 1 + (-0.548 + 0.836i)T \)
good2 \( 1 + (-0.533 - 0.404i)T + (0.269 + 0.962i)T^{2} \)
7 \( 1 + (0.136 + 0.990i)T^{2} \)
11 \( 1 + (-0.0682 + 0.997i)T^{2} \)
13 \( 1 + (-0.942 + 0.334i)T^{2} \)
17 \( 1 + (0.136 - 0.00465i)T + (0.997 - 0.0682i)T^{2} \)
19 \( 1 + (0.180 + 0.508i)T + (-0.775 + 0.631i)T^{2} \)
23 \( 1 + (0.164 + 0.216i)T + (-0.269 + 0.962i)T^{2} \)
29 \( 1 + (0.334 - 0.942i)T^{2} \)
31 \( 1 + (-0.759 - 1.24i)T + (-0.460 + 0.887i)T^{2} \)
37 \( 1 + (-0.398 + 0.917i)T^{2} \)
41 \( 1 + (0.682 + 0.730i)T^{2} \)
43 \( 1 + (-0.519 - 0.854i)T^{2} \)
53 \( 1 + (0.855 - 0.337i)T + (0.730 - 0.682i)T^{2} \)
59 \( 1 + (0.854 + 0.519i)T^{2} \)
61 \( 1 + (-0.234 - 1.12i)T + (-0.917 + 0.398i)T^{2} \)
67 \( 1 + (0.136 - 0.990i)T^{2} \)
71 \( 1 + (-0.962 - 0.269i)T^{2} \)
73 \( 1 + (-0.816 + 0.576i)T^{2} \)
79 \( 1 + (-0.861 - 1.05i)T + (-0.203 + 0.979i)T^{2} \)
83 \( 1 + (1.83 + 0.0626i)T + (0.997 + 0.0682i)T^{2} \)
89 \( 1 + (-0.775 - 0.631i)T^{2} \)
97 \( 1 + (0.887 - 0.460i)T^{2} \)
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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

−8.690957623717041135657783609999, −7.912814621270127794634313206745, −6.95826171920677702185173931571, −6.65162768535749018956452895667, −5.65410166405939338139167181906, −4.79374971714304333451318653809, −4.12192016203019525832927371326, −3.23713613203250535956133259484, −2.21497673187398240465972415889, −1.11578384458293284780874355821, 1.71941846650990971023166653862, 2.60967266188787775330075733907, 3.30932816287861536097501613022, 4.14622431314234653480783557536, 4.62022110129315177069930094726, 5.65390038784588935779846934972, 6.67520270565190846782704150421, 7.76039173751171086240415779918, 7.940430189116544885453952654616, 8.817364542278510716853862842709

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