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Properties

Label 1-3525-3525.602-r1-0-0
Degree $1$
Conductor $3525$
Sign $-0.280 + 0.959i$
Analytic cond. $378.813$
Root an. cond. $378.813$
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  + (0.295 + 0.955i)2-s + (−0.824 + 0.565i)4-s + (0.887 + 0.460i)7-s + (−0.784 − 0.620i)8-s + (0.758 − 0.652i)11-s + (0.995 + 0.0954i)13-s + (−0.176 + 0.984i)14-s + (0.360 − 0.932i)16-s + (0.922 + 0.385i)17-s + (−0.981 − 0.190i)19-s + (0.847 + 0.531i)22-s + (0.800 − 0.598i)23-s + (0.203 + 0.979i)26-s + (−0.992 + 0.122i)28-s + (−0.976 − 0.216i)29-s + ⋯
L(s)  = 1  + (0.295 + 0.955i)2-s + (−0.824 + 0.565i)4-s + (0.887 + 0.460i)7-s + (−0.784 − 0.620i)8-s + (0.758 − 0.652i)11-s + (0.995 + 0.0954i)13-s + (−0.176 + 0.984i)14-s + (0.360 − 0.932i)16-s + (0.922 + 0.385i)17-s + (−0.981 − 0.190i)19-s + (0.847 + 0.531i)22-s + (0.800 − 0.598i)23-s + (0.203 + 0.979i)26-s + (−0.992 + 0.122i)28-s + (−0.976 − 0.216i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.280 + 0.959i$
Analytic conductor: \(378.813\)
Root analytic conductor: \(378.813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3525,\ (1:\ ),\ -0.280 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896879201 + 2.530836852i\)
\(L(\frac12)\) \(\approx\) \(1.896879201 + 2.530836852i\)
\(L(1)\) \(\approx\) \(1.179325340 + 0.7371170031i\)
\(L(1)\) \(\approx\) \(1.179325340 + 0.7371170031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.295 + 0.955i)T \)
7 \( 1 + (0.887 + 0.460i)T \)
11 \( 1 + (0.758 - 0.652i)T \)
13 \( 1 + (0.995 + 0.0954i)T \)
17 \( 1 + (0.922 + 0.385i)T \)
19 \( 1 + (-0.981 - 0.190i)T \)
23 \( 1 + (0.800 - 0.598i)T \)
29 \( 1 + (-0.976 - 0.216i)T \)
31 \( 1 + (-0.998 - 0.0546i)T \)
37 \( 1 + (0.984 - 0.176i)T \)
41 \( 1 + (0.0409 + 0.999i)T \)
43 \( 1 + (-0.942 - 0.334i)T \)
53 \( 1 + (0.784 - 0.620i)T \)
59 \( 1 + (0.792 - 0.609i)T \)
61 \( 1 + (-0.176 + 0.984i)T \)
67 \( 1 + (-0.447 + 0.894i)T \)
71 \( 1 + (-0.946 + 0.321i)T \)
73 \( 1 + (-0.109 + 0.994i)T \)
79 \( 1 + (-0.641 + 0.767i)T \)
83 \( 1 + (-0.0818 - 0.996i)T \)
89 \( 1 + (-0.484 - 0.874i)T \)
97 \( 1 + (0.0546 + 0.998i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.358825203797317496711335102334, −17.93089854955034669569377038522, −17.027485244978497005859554911119, −16.5807908321524239342018658414, −15.15917453801182498081195114425, −14.85990641139813738064591085127, −14.13153041361101348363596462252, −13.447271792139509093361907621032, −12.78570623108592000419747298697, −12.02317536033660065450656401466, −11.32418086453870365503779696970, −10.85640209767404493901559968957, −10.11707193248161174694144440960, −9.273924884452245788544923195694, −8.709675165671736074060395775571, −7.80954611996211508758286319676, −7.016671752233933733980188976322, −5.94726710033347074835193883772, −5.28095929088513850911182915314, −4.41920133242709463024343377259, −3.84407510859565879237134051687, −3.12376627452702994323208525226, −1.86253513686225317226999456147, −1.486806092337684086890562828, −0.56958810416825938598461885172, 0.755299888307659819469173699480, 1.63403599510851781638194808309, 2.86055498609902463656500325617, 3.80590506915753708391177025879, 4.31252230840100310305827520502, 5.35591095112728479392475052173, 5.85245210725339927341750417681, 6.55423473888242822620139010977, 7.3876019331676364051082191161, 8.28920963382551656587067561790, 8.64075114482778690548679841432, 9.284715055716308548324575579029, 10.36947159987556211089284826936, 11.33342594493987305671877995103, 11.72542945560094223748904062748, 12.89130687155801316252931176578, 13.16778130929344497357880229386, 14.230099811801162740707265088829, 14.7483068202525552694573982191, 15.04868794597519868111546804505, 16.13882610261755326634899205823, 16.67087521904820066349598203843, 17.176935083520711195373925251666, 18.03760592052096220732123049707, 18.65127736541093852586092663345

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