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Properties

Label 2-525-5.2-c2-0-25
Degree $2$
Conductor $525$
Sign $0.850 + 0.525i$
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  + (1.36 + 1.36i)2-s + (1.22 − 1.22i)3-s − 0.258i·4-s + 3.35·6-s + (−1.87 − 1.87i)7-s + (5.82 − 5.82i)8-s − 2.99i·9-s + 17.6·11-s + (−0.316 − 0.316i)12-s + (−12.1 + 12.1i)13-s − 5.11i·14-s + 14.9·16-s + (−13.8 − 13.8i)17-s + (4.10 − 4.10i)18-s − 18.3i·19-s + ⋯
L(s)  = 1  + (0.683 + 0.683i)2-s + (0.408 − 0.408i)3-s − 0.0645i·4-s + 0.558·6-s + (−0.267 − 0.267i)7-s + (0.728 − 0.728i)8-s − 0.333i·9-s + 1.60·11-s + (−0.0263 − 0.0263i)12-s + (−0.932 + 0.932i)13-s − 0.365i·14-s + 0.931·16-s + (−0.816 − 0.816i)17-s + (0.227 − 0.227i)18-s − 0.963i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.995887988\)
\(L(\frac12)\) \(\approx\) \(2.995887988\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (-1.36 - 1.36i)T + 4iT^{2} \)
11 \( 1 - 17.6T + 121T^{2} \)
13 \( 1 + (12.1 - 12.1i)T - 169iT^{2} \)
17 \( 1 + (13.8 + 13.8i)T + 289iT^{2} \)
19 \( 1 + 18.3iT - 361T^{2} \)
23 \( 1 + (-26.3 + 26.3i)T - 529iT^{2} \)
29 \( 1 + 2.87iT - 841T^{2} \)
31 \( 1 - 16.1T + 961T^{2} \)
37 \( 1 + (2.52 + 2.52i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.89T + 1.68e3T^{2} \)
43 \( 1 + (-42.5 + 42.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-57.7 - 57.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (66.5 - 66.5i)T - 2.80e3iT^{2} \)
59 \( 1 - 16.4iT - 3.48e3T^{2} \)
61 \( 1 + 7.37T + 3.72e3T^{2} \)
67 \( 1 + (-27.2 - 27.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 79.5T + 5.04e3T^{2} \)
73 \( 1 + (63.3 - 63.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 2.48iT - 6.24e3T^{2} \)
83 \( 1 + (-29.0 + 29.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 29.3iT - 7.92e3T^{2} \)
97 \( 1 + (-89.1 - 89.1i)T + 9.40e3iT^{2} \)
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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

−10.59995482608538168717803560836, −9.264751406640516137264896173177, −9.080256307003687570881971434105, −7.33713791014299242552391192891, −6.88111368460462778196156643417, −6.23647628236781321378917526706, −4.73977276095530227068040118643, −4.16913540888890784421326190285, −2.58047340266106135411183493029, −0.982706669934891377476772670429, 1.71845138282490313848857297580, 3.01804060414010534288260687004, 3.79959952787488976187876756705, 4.72738682055359778462564689537, 5.86580880229659743725715414052, 7.14777292683774513714567935024, 8.168099752657849500775390325858, 9.064970820388890338785860438031, 9.930303959773592458264876741482, 10.87821537930451686126634944448

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