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Properties

Label 2-525-35.4-c1-0-13
Degree $2$
Conductor $525$
Sign $0.997 - 0.0667i$
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  + (−0.866 − 0.5i)3-s + (−1 + 1.73i)4-s + (0.866 − 2.5i)7-s + (0.499 + 0.866i)9-s + (1.73 − 0.999i)12-s − i·13-s + (−1.99 − 3.46i)16-s + (5.19 + 3i)17-s + (2.5 + 4.33i)19-s + (−2 + 1.73i)21-s + (5.19 − 3i)23-s − 0.999i·27-s + (3.46 + 4i)28-s + 6·29-s + (−2.5 + 4.33i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (−0.5 + 0.866i)4-s + (0.327 − 0.944i)7-s + (0.166 + 0.288i)9-s + (0.499 − 0.288i)12-s − 0.277i·13-s + (−0.499 − 0.866i)16-s + (1.26 + 0.727i)17-s + (0.573 + 0.993i)19-s + (−0.436 + 0.377i)21-s + (1.08 − 0.625i)23-s − 0.192i·27-s + (0.654 + 0.755i)28-s + 1.11·29-s + (−0.449 + 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.997 - 0.0667i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.997 - 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16097 + 0.0387906i\)
\(L(\frac12)\) \(\approx\) \(1.16097 + 0.0387906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.866 + 2.5i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (5.19 - 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (9.52 + 5.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.82682461350037902337995881371, −10.15515767739284118952155696436, −9.021573996399443835918117483807, −7.84075259522825185955111820836, −7.58597406066573915460401618623, −6.32801687757455377943876200125, −5.13611756408748007217213322032, −4.15772028007840777377121972315, −3.11217103927366761642988700715, −1.07493006068174258266583642518, 1.06932122275175570233074872956, 2.83645989017408715754363012947, 4.50102697823097930090977823458, 5.27490307712484451120293809952, 5.92284392863499565909039897782, 7.09993691160973803001616038346, 8.349737305008646119189790458382, 9.522134894010389941524533352420, 9.606981373993867686643712257016, 11.01883944715871003591154350337

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