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Properties

Label 2-575-5.4-c3-0-6
Degree $2$
Conductor $575$
Sign $0.447 + 0.894i$
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  − 4.96i·2-s − 7.15i·3-s − 16.6·4-s − 35.5·6-s + 3.69i·7-s + 42.7i·8-s − 24.2·9-s − 25.2·11-s + 118. i·12-s + 29.3i·13-s + 18.3·14-s + 79.1·16-s + 80.9i·17-s + 120. i·18-s + 48.2·19-s + ⋯
L(s)  = 1  − 1.75i·2-s − 1.37i·3-s − 2.07·4-s − 2.41·6-s + 0.199i·7-s + 1.88i·8-s − 0.896·9-s − 0.693·11-s + 2.86i·12-s + 0.625i·13-s + 0.349·14-s + 1.23·16-s + 1.15i·17-s + 1.57i·18-s + 0.582·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(33.9260\)
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: \((2,\ 575,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6656362144\)
\(L(\frac12)\) \(\approx\) \(0.6656362144\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.96iT - 8T^{2} \)
3 \( 1 + 7.15iT - 27T^{2} \)
7 \( 1 - 3.69iT - 343T^{2} \)
11 \( 1 + 25.2T + 1.33e3T^{2} \)
13 \( 1 - 29.3iT - 2.19e3T^{2} \)
17 \( 1 - 80.9iT - 4.91e3T^{2} \)
19 \( 1 - 48.2T + 6.85e3T^{2} \)
29 \( 1 - 161.T + 2.43e4T^{2} \)
31 \( 1 + 261.T + 2.97e4T^{2} \)
37 \( 1 + 37.1iT - 5.06e4T^{2} \)
41 \( 1 + 135.T + 6.89e4T^{2} \)
43 \( 1 - 304. iT - 7.95e4T^{2} \)
47 \( 1 - 200. iT - 1.03e5T^{2} \)
53 \( 1 + 384. iT - 1.48e5T^{2} \)
59 \( 1 + 113.T + 2.05e5T^{2} \)
61 \( 1 + 763.T + 2.26e5T^{2} \)
67 \( 1 - 736. iT - 3.00e5T^{2} \)
71 \( 1 - 721.T + 3.57e5T^{2} \)
73 \( 1 + 380. iT - 3.89e5T^{2} \)
79 \( 1 + 754.T + 4.93e5T^{2} \)
83 \( 1 - 1.18e3iT - 5.71e5T^{2} \)
89 \( 1 + 611.T + 7.04e5T^{2} \)
97 \( 1 + 371. iT - 9.12e5T^{2} \)
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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

−10.42052993829885563929255167562, −9.469211873940126044313536256686, −8.537285549005033450547886682533, −7.70716679459698732726672511399, −6.57943958333695508612400252045, −5.39100374557591419276929767613, −4.11021936565182095476877787367, −2.90722680952208905230533158755, −1.95749975864769479801103215298, −1.16487393682732829392293628168, 0.21999693834752914058389280658, 3.12845530579733731621139992706, 4.31190681461206794674446533583, 5.12930830513413430404681391330, 5.64405286598408971864614133597, 6.97679627404191382927944118205, 7.68875365203034957317047536212, 8.689210889859432384517035660276, 9.382268620349151378065105768773, 10.17850000694931834986363889509

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