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Properties

Label 2-525-35.24-c2-0-35
Degree $2$
Conductor $525$
Sign $0.652 + 0.758i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 − 1.5i)2-s + (−0.866 + 1.5i)3-s + (2.5 − 4.33i)4-s + 5.19i·6-s + (6.06 − 3.5i)7-s − 3.00i·8-s + (−1.5 − 2.59i)9-s + (4.33 + 7.5i)12-s + 3.46·13-s + (10.5 − 18.1i)14-s + (5.49 + 9.52i)16-s + (7.79 − 13.5i)17-s + (−7.79 − 4.5i)18-s + (24 − 13.8i)19-s + 12.1i·21-s + ⋯
L(s)  = 1  + (1.29 − 0.750i)2-s + (−0.288 + 0.5i)3-s + (0.625 − 1.08i)4-s + 0.866i·6-s + (0.866 − 0.5i)7-s − 0.375i·8-s + (−0.166 − 0.288i)9-s + (0.360 + 0.625i)12-s + 0.266·13-s + (0.750 − 1.29i)14-s + (0.343 + 0.595i)16-s + (0.458 − 0.794i)17-s + (−0.433 − 0.250i)18-s + (1.26 − 0.729i)19-s + 0.577i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.652 + 0.758i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.652 + 0.758i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.679242132\)
\(L(\frac12)\) \(\approx\) \(3.679242132\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-6.06 + 3.5i)T \)
good2 \( 1 + (-2.59 + 1.5i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 3.46T + 169T^{2} \)
17 \( 1 + (-7.79 + 13.5i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-24 + 13.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.9 + 7.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 6T + 841T^{2} \)
31 \( 1 + (-19.5 - 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (60.6 - 35i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 36.3iT - 1.68e3T^{2} \)
43 \( 1 - 34iT - 1.84e3T^{2} \)
47 \( 1 + (-12.9 - 22.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (36.3 + 21i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (36 + 20.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-63 + 36.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (81.4 + 47i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 9T + 5.04e3T^{2} \)
73 \( 1 + (-3.46 + 6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-38.5 - 66.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 145.T + 6.88e3T^{2} \)
89 \( 1 + (49.5 - 28.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 98.7T + 9.40e3T^{2} \)
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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

−10.85508941572149268411568726471, −10.06047320298952218241318846365, −8.914092868358120014410543798473, −7.74238310678457502865774426150, −6.58707564048309617788210944968, −5.16577600018367567833224361167, −4.96252082785889296498155795079, −3.77400915939293341599789398329, −2.83928672970195085903151881573, −1.23252833976567467564242824834, 1.48167969001001100221929291094, 3.14541816152167070867027225674, 4.31527718365009102169297420455, 5.48279359194409316047714355948, 5.76501443292481956653696198689, 7.00783194025909426909849541360, 7.72333100014800725364203670160, 8.640458688866510235456331659436, 9.981846410505965233802914984200, 11.13858938876875665517901507858

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