Properties

Label 2-575-5.4-c1-0-8
Degree $2$
Conductor $575$
Sign $-0.894 - 0.447i$
Analytic cond. $4.59139$
Root an. cond. $2.14275$
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  + 2i·2-s − 2·4-s + i·7-s + 3·9-s + 2·11-s + 2i·13-s − 2·14-s − 4·16-s + 3i·17-s + 6i·18-s + 2·19-s + 4i·22-s i·23-s − 4·26-s − 2i·28-s − 7·29-s + ⋯
L(s)  = 1  + 1.41i·2-s − 4-s + 0.377i·7-s + 9-s + 0.603·11-s + 0.554i·13-s − 0.534·14-s − 16-s + 0.727i·17-s + 1.41i·18-s + 0.458·19-s + 0.852i·22-s − 0.208i·23-s − 0.784·26-s − 0.377i·28-s − 1.29·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.346099 + 1.46610i\)
\(L(\frac12)\) \(\approx\) \(0.346099 + 1.46610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 + iT \)
good2 \( 1 - 2iT - 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 11iT - 37T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 11iT - 53T^{2} \)
59 \( 1 - 13T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−11.12603106909784915888462972923, −9.911650470255905452998073991935, −9.117385913710690676553716833466, −8.294550127333029008521408761706, −7.32954218148021851946669290725, −6.69641153506606733018488548265, −5.82314546591211313986990596501, −4.82147163451480733762566181160, −3.78566970356809074661979933285, −1.84924426083646621034340674583, 0.927667241196053907224812344189, 2.15998555641500017668091684390, 3.53081791084325619241452337711, 4.19947663910706765395402460145, 5.47933197517217055246150379731, 6.94880234923996874846823332893, 7.59551044214332695348181727113, 9.188243279250411471900586394421, 9.560933495225478734515104175588, 10.57640785131400796827869400117

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