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Properties

Label 2-525-7.2-c1-0-16
Degree $2$
Conductor $525$
Sign $0.991 + 0.126i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (0.500 + 0.866i)4-s + 0.999·6-s + (0.5 − 2.59i)7-s + 3·8-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + 3·13-s + (−2 − 1.73i)14-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s + (2.5 − 0.866i)21-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 + 0.433i)4-s + 0.408·6-s + (0.188 − 0.981i)7-s + 1.06·8-s + (−0.166 + 0.288i)9-s + (−0.144 + 0.249i)12-s + 0.832·13-s + (−0.534 − 0.462i)14-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + (0.117 + 0.204i)18-s + (−0.114 + 0.198i)19-s + (0.545 − 0.188i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20892 - 0.140178i\)
\(L(\frac12)\) \(\approx\) \(2.20892 - 0.140178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7 + 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9T + 97T^{2} \)
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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

−10.85566093461833198946429671344, −10.31387055716331359597992124270, −9.154941588932228164117478680324, −8.088390061054786499721771312548, −7.42326257301075120959504043773, −6.22736336753547010862975341965, −4.75283830866069395956696057002, −3.91477667406112519572011350224, −3.14320662080991212618876856284, −1.61071153691613638870898303207, 1.50776992908578175219466228173, 2.78385899371179767270100688397, 4.37920712784357924805484873752, 5.63697526841829417306955217803, 6.10049495131599517606659189673, 7.20902746717682161342392885518, 8.002953728896567939327330086133, 8.991140583676504853180947362312, 9.858039432859478730609667263865, 11.12145146344478155155220010411

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