↑

Properties

Label 2-575-115.29-c1-0-2
Degree $2$
Conductor $575$
Sign $0.120 - 0.992i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.02 − 0.925i)2-s + (−0.281 + 0.959i)3-s + (1.94 + 2.24i)4-s + (1.45 − 1.68i)6-s + (0.0872 + 0.0125i)7-s + (−0.607 − 2.06i)8-s + (1.68 + 1.08i)9-s + (1.01 + 2.21i)11-s + (−2.69 + 1.23i)12-s + (−3.56 + 0.512i)13-s + (−0.165 − 0.106i)14-s + (0.160 − 1.11i)16-s + (2.55 + 2.21i)17-s + (−2.40 − 3.74i)18-s + (−1.87 − 2.16i)19-s + ⋯
L(s)  = 1  + (−1.43 − 0.654i)2-s + (−0.162 + 0.553i)3-s + (0.971 + 1.12i)4-s + (0.595 − 0.687i)6-s + (0.0329 + 0.00474i)7-s + (−0.214 − 0.731i)8-s + (0.560 + 0.360i)9-s + (0.305 + 0.669i)11-s + (−0.779 + 0.355i)12-s + (−0.988 + 0.142i)13-s + (−0.0441 − 0.0283i)14-s + (0.0402 − 0.279i)16-s + (0.620 + 0.537i)17-s + (−0.568 − 0.883i)18-s + (−0.429 − 0.496i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $0.120 - 0.992i$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ 0.120 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.389237 + 0.344832i\)
\(L(\frac12)\) \(\approx\) \(0.389237 + 0.344832i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 + 4.28i)T \)
good2 \( 1 + (2.02 + 0.925i)T + (1.30 + 1.51i)T^{2} \)
3 \( 1 + (0.281 - 0.959i)T + (-2.52 - 1.62i)T^{2} \)
7 \( 1 + (-0.0872 - 0.0125i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (-1.01 - 2.21i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (3.56 - 0.512i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-2.55 - 2.21i)T + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.87 + 2.16i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (4.87 - 5.62i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-1.65 + 0.485i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-0.382 + 0.595i)T + (-15.3 - 33.6i)T^{2} \)
41 \( 1 + (4.66 - 3.00i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (0.675 - 2.30i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + (-10.2 - 1.47i)T + (50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.64 - 11.4i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (3.89 - 1.14i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-11.1 - 5.07i)T + (43.8 + 50.6i)T^{2} \)
71 \( 1 + (-0.787 + 1.72i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (4.16 - 3.61i)T + (10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.997 - 6.93i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.25 - 3.51i)T + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (15.3 + 4.49i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (7.77 + 12.0i)T + (-40.2 + 88.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64426112315596014580799491029, −10.00667269399527044793969839467, −9.443251419928465471384620069539, −8.559547881162993319385661602014, −7.56358118242801264807883819419, −6.86676244567740513867063208276, −5.20794816715481955101010520459, −4.24546120510149155018624425470, −2.70045468147768562505291248993, −1.49909094777513155010171442308, 0.50532733697076014473038353560, 1.85048901650731965428755662109, 3.69700244764505996029891056590, 5.36738339023161694636176292638, 6.38839054686969701421060698470, 7.17345963533746745013541490867, 7.78223813419543063137929692362, 8.670377490306049075809490587863, 9.686324286176476641862418973839, 10.03848396316697324418465829748

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