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Properties

Label 2-3525-705.407-c0-0-0
Degree $2$
Conductor $3525$
Sign $0.307 - 0.951i$
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  + (0.395 − 0.0963i)2-s + (0.169 − 0.985i)3-s + (−0.740 + 0.383i)4-s + (−0.0277 − 0.405i)6-s + (−0.562 + 0.490i)8-s + (−0.942 − 0.334i)9-s + (0.252 + 0.795i)12-s + (0.306 − 0.433i)16-s + (−1.53 + 1.16i)17-s + (−0.404 − 0.0416i)18-s + (0.361 + 1.73i)19-s + (−0.246 + 1.00i)23-s + (0.387 + 0.637i)24-s + (−0.490 + 0.871i)27-s + (−0.222 − 0.157i)31-s + (0.353 − 0.895i)32-s + ⋯
L(s)  = 1  + (0.395 − 0.0963i)2-s + (0.169 − 0.985i)3-s + (−0.740 + 0.383i)4-s + (−0.0277 − 0.405i)6-s + (−0.562 + 0.490i)8-s + (−0.942 − 0.334i)9-s + (0.252 + 0.795i)12-s + (0.306 − 0.433i)16-s + (−1.53 + 1.16i)17-s + (−0.404 − 0.0416i)18-s + (0.361 + 1.73i)19-s + (−0.246 + 1.00i)23-s + (0.387 + 0.637i)24-s + (−0.490 + 0.871i)27-s + (−0.222 − 0.157i)31-s + (0.353 − 0.895i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.307 - 0.951i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7579087561\)
\(L(\frac12)\) \(\approx\) \(0.7579087561\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.169 + 0.985i)T \)
5 \( 1 \)
47 \( 1 + (0.999 + 0.0341i)T \)
good2 \( 1 + (-0.395 + 0.0963i)T + (0.887 - 0.460i)T^{2} \)
7 \( 1 + (-0.519 + 0.854i)T^{2} \)
11 \( 1 + (0.962 - 0.269i)T^{2} \)
13 \( 1 + (-0.979 + 0.203i)T^{2} \)
17 \( 1 + (1.53 - 1.16i)T + (0.269 - 0.962i)T^{2} \)
19 \( 1 + (-0.361 - 1.73i)T + (-0.917 + 0.398i)T^{2} \)
23 \( 1 + (0.246 - 1.00i)T + (-0.887 - 0.460i)T^{2} \)
29 \( 1 + (-0.203 + 0.979i)T^{2} \)
31 \( 1 + (0.222 + 0.157i)T + (0.334 + 0.942i)T^{2} \)
37 \( 1 + (0.997 - 0.0682i)T^{2} \)
41 \( 1 + (-0.990 + 0.136i)T^{2} \)
43 \( 1 + (-0.816 - 0.576i)T^{2} \)
53 \( 1 + (0.440 - 0.504i)T + (-0.136 - 0.990i)T^{2} \)
59 \( 1 + (-0.576 - 0.816i)T^{2} \)
61 \( 1 + (-1.05 - 1.13i)T + (-0.0682 + 0.997i)T^{2} \)
67 \( 1 + (-0.519 - 0.854i)T^{2} \)
71 \( 1 + (-0.460 + 0.887i)T^{2} \)
73 \( 1 + (-0.631 - 0.775i)T^{2} \)
79 \( 1 + (-0.789 - 1.81i)T + (-0.682 + 0.730i)T^{2} \)
83 \( 1 + (-0.108 - 0.0824i)T + (0.269 + 0.962i)T^{2} \)
89 \( 1 + (-0.917 - 0.398i)T^{2} \)
97 \( 1 + (-0.942 - 0.334i)T^{2} \)
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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

−8.666516169057763204561460912810, −8.177544040648378398969246336046, −7.56691720210312363763874595143, −6.61118809325111498397874117354, −5.88544718215530773123815311346, −5.24432732964846537299426239644, −4.03191994293298664699060433109, −3.58047193858094326076107979782, −2.43331711066189936780406557079, −1.49381038000232064468141466476, 0.38108761677271708883670267399, 2.37990613966496875412074544615, 3.19700042351806160167904821711, 4.22462794852983696724400307594, 4.79547430582441247529827294405, 5.18704402352793373096610182028, 6.29416196016543235624647775726, 6.90656877656718819795703325112, 8.088881764177822200606746145115, 8.959719296691660210625964055448

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