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Properties

Label 2-525-105.2-c1-0-19
Degree $2$
Conductor $525$
Sign $0.987 + 0.154i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.243 − 0.907i)2-s + (1.31 + 1.12i)3-s + (0.967 − 0.558i)4-s + (0.700 − 1.46i)6-s + (−0.0144 + 2.64i)7-s + (−2.07 − 2.07i)8-s + (0.470 + 2.96i)9-s + (0.630 − 0.363i)11-s + (1.90 + 0.352i)12-s + (1.44 − 1.44i)13-s + (2.40 − 0.630i)14-s + (−0.257 + 0.446i)16-s + (7.09 + 1.90i)17-s + (2.57 − 1.14i)18-s + (−0.664 − 0.383i)19-s + ⋯
L(s)  = 1  + (−0.171 − 0.641i)2-s + (0.760 + 0.649i)3-s + (0.483 − 0.279i)4-s + (0.285 − 0.599i)6-s + (−0.00544 + 0.999i)7-s + (−0.732 − 0.732i)8-s + (0.156 + 0.987i)9-s + (0.189 − 0.109i)11-s + (0.549 + 0.101i)12-s + (0.400 − 0.400i)13-s + (0.642 − 0.168i)14-s + (−0.0644 + 0.111i)16-s + (1.71 + 0.460i)17-s + (0.606 − 0.270i)18-s + (−0.152 − 0.0879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.987 + 0.154i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97207 - 0.153235i\)
\(L(\frac12)\) \(\approx\) \(1.97207 - 0.153235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.31 - 1.12i)T \)
5 \( 1 \)
7 \( 1 + (0.0144 - 2.64i)T \)
good2 \( 1 + (0.243 + 0.907i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.630 + 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 + 1.44i)T - 13iT^{2} \)
17 \( 1 + (-7.09 - 1.90i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.664 + 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.13 + 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.08 - 1.63i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (-5.15 + 5.15i)T - 43iT^{2} \)
47 \( 1 + (1.82 + 6.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.41 - 5.26i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.807 + 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.84 - 6.90i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.06iT - 71T^{2} \)
73 \( 1 + (15.2 + 4.08i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.80 + 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.83 - 1.83i)T + 83iT^{2} \)
89 \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.62 + 5.62i)T + 97iT^{2} \)
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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

−10.66887772748399086617192794785, −10.03020087516672712568207250024, −9.202163244001124435731275228726, −8.468710814381047676535046045546, −7.41189924956816145500009886191, −6.04020544848710685927174282673, −5.23704520597510426926251813436, −3.61278518016462977453053367227, −2.89010385516660062513637117490, −1.66290334608743042693970054003, 1.40543017415460975488702811497, 2.96167646709209181221855929891, 3.88999743114081569312102071247, 5.59962671568100710630474976955, 6.69132464258661929053913657229, 7.34767939651191929338132252288, 7.899142561060605263031672366973, 8.875936975677840881648951168568, 9.771289496070834049708664258757, 10.92912979572219363315056660955

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