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Properties

Label 2-3525-705.203-c0-0-0
Degree $2$
Conductor $3525$
Sign $-0.991 - 0.127i$
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  + (0.202 + 1.97i)2-s + (−0.930 + 0.366i)3-s + (−2.86 + 0.595i)4-s + (−0.911 − 1.75i)6-s + (−1.15 − 3.63i)8-s + (0.730 − 0.682i)9-s + (2.44 − 1.60i)12-s + (4.25 − 1.84i)16-s + (−0.113 + 0.660i)17-s + (1.49 + 1.30i)18-s + (1.93 + 0.266i)19-s + (1.25 + 0.129i)23-s + (2.40 + 2.95i)24-s + (−0.429 + 0.903i)27-s + (−0.650 − 1.49i)31-s + (2.63 + 4.67i)32-s + ⋯
L(s)  = 1  + (0.202 + 1.97i)2-s + (−0.930 + 0.366i)3-s + (−2.86 + 0.595i)4-s + (−0.911 − 1.75i)6-s + (−1.15 − 3.63i)8-s + (0.730 − 0.682i)9-s + (2.44 − 1.60i)12-s + (4.25 − 1.84i)16-s + (−0.113 + 0.660i)17-s + (1.49 + 1.30i)18-s + (1.93 + 0.266i)19-s + (1.25 + 0.129i)23-s + (2.40 + 2.95i)24-s + (−0.429 + 0.903i)27-s + (−0.650 − 1.49i)31-s + (2.63 + 4.67i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.991 - 0.127i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8375186174\)
\(L(\frac12)\) \(\approx\) \(0.8375186174\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.930 - 0.366i)T \)
5 \( 1 \)
47 \( 1 + (0.971 + 0.236i)T \)
good2 \( 1 + (-0.202 - 1.97i)T + (-0.979 + 0.203i)T^{2} \)
7 \( 1 + (-0.631 + 0.775i)T^{2} \)
11 \( 1 + (-0.334 - 0.942i)T^{2} \)
13 \( 1 + (-0.136 + 0.990i)T^{2} \)
17 \( 1 + (0.113 - 0.660i)T + (-0.942 - 0.334i)T^{2} \)
19 \( 1 + (-1.93 - 0.266i)T + (0.962 + 0.269i)T^{2} \)
23 \( 1 + (-1.25 - 0.129i)T + (0.979 + 0.203i)T^{2} \)
29 \( 1 + (0.990 - 0.136i)T^{2} \)
31 \( 1 + (0.650 + 1.49i)T + (-0.682 + 0.730i)T^{2} \)
37 \( 1 + (0.887 - 0.460i)T^{2} \)
41 \( 1 + (-0.576 + 0.816i)T^{2} \)
43 \( 1 + (0.398 + 0.917i)T^{2} \)
53 \( 1 + (-1.30 - 0.412i)T + (0.816 + 0.576i)T^{2} \)
59 \( 1 + (-0.917 - 0.398i)T^{2} \)
61 \( 1 + (-0.116 + 0.0709i)T + (0.460 - 0.887i)T^{2} \)
67 \( 1 + (-0.631 - 0.775i)T^{2} \)
71 \( 1 + (-0.203 + 0.979i)T^{2} \)
73 \( 1 + (-0.997 + 0.0682i)T^{2} \)
79 \( 1 + (0.311 - 1.11i)T + (-0.854 - 0.519i)T^{2} \)
83 \( 1 + (-0.156 - 0.906i)T + (-0.942 + 0.334i)T^{2} \)
89 \( 1 + (0.962 - 0.269i)T^{2} \)
97 \( 1 + (0.730 - 0.682i)T^{2} \)
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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

−8.988457298031974891593401596531, −8.121394538021442035367754441448, −7.30387451168100807339282392834, −6.93740366403288899681260876270, −5.99478199291098093135392484057, −5.51133709458929931935819644770, −4.96126974280160786243975151911, −4.06555976832186340605114859470, −3.40164370008083773927929627452, −0.926411135918819361806622702022, 0.803611486932574751570089729054, 1.59396389375125726128275501227, 2.76907961259388126470488777140, 3.43396975895255122359845599298, 4.57846295492353224426761345355, 5.12740143354736529034858047901, 5.64687430565459269113726972140, 6.92910480181636774626699905462, 7.72808062446952253316142053123, 8.842851469044339551240303892149

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