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Properties

Label 2-575-5.4-c3-0-12
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  + 4.39i·2-s + 3.27i·3-s − 11.2·4-s − 14.4·6-s + 13.7i·7-s − 14.4i·8-s + 16.2·9-s − 16.7·11-s − 37.0i·12-s + 68.7i·13-s − 60.5·14-s − 26.8·16-s − 40.7i·17-s + 71.3i·18-s − 40.6·19-s + ⋯
L(s)  = 1  + 1.55i·2-s + 0.631i·3-s − 1.41·4-s − 0.979·6-s + 0.745i·7-s − 0.638i·8-s + 0.601·9-s − 0.457·11-s − 0.890i·12-s + 1.46i·13-s − 1.15·14-s − 0.419·16-s − 0.580i·17-s + 0.934i·18-s − 0.490·19-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

\(L(2)\) \(\approx\) \(1.022136663\)
\(L(\frac12)\) \(\approx\) \(1.022136663\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.39iT - 8T^{2} \)
3 \( 1 - 3.27iT - 27T^{2} \)
7 \( 1 - 13.7iT - 343T^{2} \)
11 \( 1 + 16.7T + 1.33e3T^{2} \)
13 \( 1 - 68.7iT - 2.19e3T^{2} \)
17 \( 1 + 40.7iT - 4.91e3T^{2} \)
19 \( 1 + 40.6T + 6.85e3T^{2} \)
29 \( 1 + 41.4T + 2.43e4T^{2} \)
31 \( 1 + 234.T + 2.97e4T^{2} \)
37 \( 1 - 101. iT - 5.06e4T^{2} \)
41 \( 1 - 9.09T + 6.89e4T^{2} \)
43 \( 1 - 146. iT - 7.95e4T^{2} \)
47 \( 1 + 383. iT - 1.03e5T^{2} \)
53 \( 1 + 430. iT - 1.48e5T^{2} \)
59 \( 1 - 441.T + 2.05e5T^{2} \)
61 \( 1 - 73.6T + 2.26e5T^{2} \)
67 \( 1 + 245. iT - 3.00e5T^{2} \)
71 \( 1 + 241.T + 3.57e5T^{2} \)
73 \( 1 + 187. iT - 3.89e5T^{2} \)
79 \( 1 + 109.T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 + 148.T + 7.04e5T^{2} \)
97 \( 1 - 1.82e3iT - 9.12e5T^{2} \)
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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

−10.96321159383631666038191139721, −9.724106912736078328591710424400, −9.141690490582525704437449927756, −8.375284236048621979999666378696, −7.26423384399983628164616954665, −6.66868723524468225527373669463, −5.54757824943665107877105377658, −4.84801991148764843476267102006, −3.88736558825161694777186216548, −2.09981100694326362531243158657, 0.31219497767776326391822687222, 1.30508055268604303715811934840, 2.40060295552086596978302949709, 3.54926962245525140231624621211, 4.43221643620807362067709026034, 5.76851850214018035752018765633, 7.09966394200557666845536795875, 7.83649882143647053516971409206, 8.932613101500465012854452945696, 10.11923079977613830247086523706

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