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Properties

Label 2-525-35.19-c2-0-7
Degree $2$
Conductor $525$
Sign $-0.0376 - 0.999i$
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  + (0.583 + 0.336i)2-s + (−0.866 − 1.5i)3-s + (−1.77 − 3.07i)4-s − 1.16i·6-s + (1.55 + 6.82i)7-s − 5.08i·8-s + (−1.5 + 2.59i)9-s + (0.0223 + 0.0387i)11-s + (−3.07 + 5.31i)12-s − 23.0·13-s + (−1.38 + 4.50i)14-s + (−5.38 + 9.32i)16-s + (4.71 + 8.16i)17-s + (−1.74 + 1.01i)18-s + (−0.991 − 0.572i)19-s + ⋯
L(s)  = 1  + (0.291 + 0.168i)2-s + (−0.288 − 0.5i)3-s + (−0.443 − 0.767i)4-s − 0.194i·6-s + (0.222 + 0.974i)7-s − 0.635i·8-s + (−0.166 + 0.288i)9-s + (0.00203 + 0.00352i)11-s + (−0.255 + 0.443i)12-s − 1.76·13-s + (−0.0992 + 0.321i)14-s + (−0.336 + 0.582i)16-s + (0.277 + 0.480i)17-s + (−0.0972 + 0.0561i)18-s + (−0.0521 − 0.0301i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0376 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0376 - 0.999i$
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: \((2,\ 525,\ (\ :1),\ -0.0376 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7947660204\)
\(L(\frac12)\) \(\approx\) \(0.7947660204\)
\(L(2)\) 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 \( 1 + (-1.55 - 6.82i)T \)
good2 \( 1 + (-0.583 - 0.336i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (-0.0223 - 0.0387i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 23.0T + 169T^{2} \)
17 \( 1 + (-4.71 - 8.16i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (0.991 + 0.572i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-38.3 - 22.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 53.0T + 841T^{2} \)
31 \( 1 + (-19.5 + 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-36.6 - 21.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 38.2iT - 1.68e3T^{2} \)
43 \( 1 - 76.5iT - 1.84e3T^{2} \)
47 \( 1 + (13.5 - 23.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-16.4 + 9.49i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-4.21 + 2.43i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.6 + 19.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.06 + 3.50i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 46.8T + 5.04e3T^{2} \)
73 \( 1 + (-41.7 - 72.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.2 + 17.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 125.T + 6.88e3T^{2} \)
89 \( 1 + (40.4 + 23.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 3.11T + 9.40e3T^{2} \)
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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

−11.06220670568745372346854511820, −9.714314521619696690238354093705, −9.386202641247569700292624966623, −8.091965939581190193309769812610, −7.15453917421432140414901588952, −6.08471974057792647172823957379, −5.32012401048837541747525502367, −4.59683448233648364684787344857, −2.80905485727286001169291721413, −1.44549830161778842475983446120, 0.30284940152413879887315589686, 2.57199005533716220240484468818, 3.75019142092154666184032457784, 4.66719252592473786465098072462, 5.31035560441763791876266784701, 7.08044785082964983913086851223, 7.51526409948761950053480874940, 8.772762772132041324675756443599, 9.580508795410760278651570848146, 10.49526520791088991982954732394

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