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Properties

Label 2-575-115.114-c0-0-0
Degree $2$
Conductor $575$
Sign $-0.447 + 0.894i$
Analytic cond. $0.286962$
Root an. cond. $0.535688$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.87i·2-s + 1.53i·3-s − 2.53·4-s − 2.87·6-s − 2.87i·8-s − 1.34·9-s − 3.87i·12-s + 0.347i·13-s + 2.87·16-s − 2.53i·18-s + i·23-s + 4.41·24-s − 0.652·26-s − 0.532i·27-s + 1.87·29-s + ⋯
L(s)  = 1  + 1.87i·2-s + 1.53i·3-s − 2.53·4-s − 2.87·6-s − 2.87i·8-s − 1.34·9-s − 3.87i·12-s + 0.347i·13-s + 2.87·16-s − 2.53i·18-s + i·23-s + 4.41·24-s − 0.652·26-s − 0.532i·27-s + 1.87·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.286962\)
Root analytic conductor: \(0.535688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7219667267\)
\(L(\frac12)\) \(\approx\) \(0.7219667267\)
\(L(1)\) 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 \( 1 - 1.87iT - T^{2} \)
3 \( 1 - 1.53iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 0.347iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 - 1.87T + T^{2} \)
31 \( 1 + 1.87T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.53T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 0.347iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 0.347T + T^{2} \)
73 \( 1 + 1.87iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.25558563259285639892666354981, −10.23629266798945432412651271668, −9.438241645335111840071140406355, −8.934687001831616528898839411906, −7.988232390931739867983474308825, −7.01743607483529399367250663896, −5.96228560171014527295483384839, −5.15856452202258393479081757602, −4.40798689412072349099126556483, −3.51092821518332235993150462880, 0.937073544916057271799528686643, 2.12927265990269035094124588639, 2.98604714065850275681710760791, 4.32658228827426446360566793189, 5.57956173745462867890786151616, 6.80813834486270783174840952236, 7.991835175420094120725930358863, 8.661583960411346231295476545303, 9.678526567071055203722415587207, 10.65076120609155294865113759515

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