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Properties

Label 2-575-5.4-c3-0-48
Degree $2$
Conductor $575$
Sign $0.894 - 0.447i$
Analytic cond. $33.9260$
Root an. cond. $5.82461$
Motivic weight $3$
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 − 5i·3-s + 4·4-s + 10·6-s + 8i·7-s + 24i·8-s + 2·9-s + 34·11-s − 20i·12-s − 57i·13-s − 16·14-s − 16·16-s + 80i·17-s + 4i·18-s + 70·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.962i·3-s + 0.5·4-s + 0.680·6-s + 0.431i·7-s + 1.06i·8-s + 0.0740·9-s + 0.931·11-s − 0.481i·12-s − 1.21i·13-s − 0.305·14-s − 0.250·16-s + 1.14i·17-s + 0.0523i·18-s + 0.845·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(33.9260\)
Root analytic conductor: \(5.82461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.735108642\)
\(L(\frac12)\) \(\approx\) \(2.735108642\)
\(L(\frac{5}{2})\) 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 \( 1 - 23iT \)
good2 \( 1 - 2iT - 8T^{2} \)
3 \( 1 + 5iT - 27T^{2} \)
7 \( 1 - 8iT - 343T^{2} \)
11 \( 1 - 34T + 1.33e3T^{2} \)
13 \( 1 + 57iT - 2.19e3T^{2} \)
17 \( 1 - 80iT - 4.91e3T^{2} \)
19 \( 1 - 70T + 6.85e3T^{2} \)
29 \( 1 + 245T + 2.43e4T^{2} \)
31 \( 1 - 103T + 2.97e4T^{2} \)
37 \( 1 - 298iT - 5.06e4T^{2} \)
41 \( 1 - 95T + 6.89e4T^{2} \)
43 \( 1 - 88iT - 7.95e4T^{2} \)
47 \( 1 - 357iT - 1.03e5T^{2} \)
53 \( 1 + 414iT - 1.48e5T^{2} \)
59 \( 1 - 408T + 2.05e5T^{2} \)
61 \( 1 - 822T + 2.26e5T^{2} \)
67 \( 1 + 926iT - 3.00e5T^{2} \)
71 \( 1 - 335T + 3.57e5T^{2} \)
73 \( 1 + 899iT - 3.89e5T^{2} \)
79 \( 1 - 1.32e3T + 4.93e5T^{2} \)
83 \( 1 + 36iT - 5.71e5T^{2} \)
89 \( 1 - 460T + 7.04e5T^{2} \)
97 \( 1 - 964iT - 9.12e5T^{2} \)
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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

−10.43749964819396897792816458512, −9.338941710641117380520438016536, −8.118822081862201329210320790935, −7.74914084702065552915789797540, −6.70696221129548232724452858336, −6.12116551003806861301770201128, −5.21855581204011814752698866323, −3.55016546655196146135037886220, −2.20327613247934851481571818358, −1.14384025185583879611576778850, 0.989664047375344293887385998698, 2.28232646954849400118117461266, 3.71157202243366749369338326063, 4.14460137014459061958063632543, 5.45411287110198440693907782189, 6.87358505207142187880071137572, 7.28356960470888393957735300881, 9.030269423578718539116638444389, 9.558971416604783464676530835586, 10.23132894454394357122530824027

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