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Properties

Label 2-525-35.19-c2-0-0
Degree $2$
Conductor $525$
Sign $-0.943 + 0.330i$
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  + (0.866 + 0.5i)2-s + (0.866 + 1.5i)3-s + (−1.5 − 2.59i)4-s + 1.73i·6-s + (−6.06 + 3.5i)7-s − 7i·8-s + (−1.5 + 2.59i)9-s + (5.5 + 9.52i)11-s + (2.59 − 4.5i)12-s − 6.92·13-s − 7·14-s + (−2.5 + 4.33i)16-s + (−12.1 − 21i)17-s + (−2.59 + 1.5i)18-s + (3 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.433 + 0.250i)2-s + (0.288 + 0.5i)3-s + (−0.375 − 0.649i)4-s + 0.288i·6-s + (−0.866 + 0.5i)7-s − 0.875i·8-s + (−0.166 + 0.288i)9-s + (0.5 + 0.866i)11-s + (0.216 − 0.375i)12-s − 0.532·13-s − 0.5·14-s + (−0.156 + 0.270i)16-s + (−0.713 − 1.23i)17-s + (−0.144 + 0.0833i)18-s + (0.157 + 0.0911i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.943 + 0.330i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05778811752\)
\(L(\frac12)\) \(\approx\) \(0.05778811752\)
\(L(2)\) 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 \( 1 + (-0.866 - 0.5i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 6.92T + 169T^{2} \)
17 \( 1 + (12.1 + 21i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (24.2 + 14i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 25T + 841T^{2} \)
31 \( 1 + (28.5 - 16.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (50.2 + 29i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 3.46iT - 1.68e3T^{2} \)
43 \( 1 - 26iT - 1.84e3T^{2} \)
47 \( 1 + (38.1 - 66i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-26.8 + 15.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-7.5 + 4.33i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-45.0 + 26i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 64T + 5.04e3T^{2} \)
73 \( 1 + (-3.46 - 6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-8.5 + 14.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 53.6T + 6.88e3T^{2} \)
89 \( 1 + (-69 - 39.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 91.7T + 9.40e3T^{2} \)
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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

−11.01988993358173219512531434351, −9.906109639951286198877793271683, −9.530419560731080373972998225447, −8.814998756056098204728915857316, −7.29308320340114391392399238007, −6.49164021669911515314159624872, −5.42197317731905775805484182602, −4.59832284147114821959911280321, −3.57966250980937068818814835443, −2.14678794659208585185344301767, 0.01770701862687644093012360215, 2.04652571637278791416313561278, 3.48084379806217800739122447766, 3.90985515857740934996793024539, 5.46659427887286999995422091137, 6.52618861226763967039356931749, 7.45962083393260797153228335721, 8.402343989556130139115303080568, 9.119571590639758575865252182235, 10.16683897395040650015711340795

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