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Properties

Label 1-575-575.259-r0-0-0
Degree $1$
Conductor $575$
Sign $0.982 + 0.183i$
Analytic cond. $2.67028$
Root an. cond. $2.67028$
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  + (−0.516 + 0.856i)2-s + (0.998 − 0.0570i)3-s + (−0.466 − 0.884i)4-s + (−0.466 + 0.884i)6-s + (0.959 + 0.281i)7-s + (0.998 + 0.0570i)8-s + (0.993 − 0.113i)9-s + (−0.921 − 0.389i)11-s + (−0.516 − 0.856i)12-s + (0.0285 − 0.999i)13-s + (−0.736 + 0.676i)14-s + (−0.564 + 0.825i)16-s + (0.466 − 0.884i)17-s + (−0.415 + 0.909i)18-s + (0.696 − 0.717i)19-s + ⋯
L(s)  = 1  + (−0.516 + 0.856i)2-s + (0.998 − 0.0570i)3-s + (−0.466 − 0.884i)4-s + (−0.466 + 0.884i)6-s + (0.959 + 0.281i)7-s + (0.998 + 0.0570i)8-s + (0.993 − 0.113i)9-s + (−0.921 − 0.389i)11-s + (−0.516 − 0.856i)12-s + (0.0285 − 0.999i)13-s + (−0.736 + 0.676i)14-s + (−0.564 + 0.825i)16-s + (0.466 − 0.884i)17-s + (−0.415 + 0.909i)18-s + (0.696 − 0.717i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $0.982 + 0.183i$
Analytic conductor: \(2.67028\)
Root analytic conductor: \(2.67028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 575,\ (0:\ ),\ 0.982 + 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594525710 + 0.1479117203i\)
\(L(\frac12)\) \(\approx\) \(1.594525710 + 0.1479117203i\)
\(L(1)\) \(\approx\) \(1.196260261 + 0.2315861479i\)
\(L(1)\) \(\approx\) \(1.196260261 + 0.2315861479i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.516 + 0.856i)T \)
3 \( 1 + (0.998 - 0.0570i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (-0.921 - 0.389i)T \)
13 \( 1 + (0.0285 - 0.999i)T \)
17 \( 1 + (0.466 - 0.884i)T \)
19 \( 1 + (0.696 - 0.717i)T \)
29 \( 1 + (0.696 + 0.717i)T \)
31 \( 1 + (-0.998 - 0.0570i)T \)
37 \( 1 + (-0.993 + 0.113i)T \)
41 \( 1 + (-0.736 - 0.676i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.941 + 0.336i)T \)
59 \( 1 + (-0.0285 + 0.999i)T \)
61 \( 1 + (0.774 + 0.633i)T \)
67 \( 1 + (-0.0855 - 0.996i)T \)
71 \( 1 + (0.974 + 0.226i)T \)
73 \( 1 + (0.985 - 0.170i)T \)
79 \( 1 + (0.610 - 0.791i)T \)
83 \( 1 + (-0.198 + 0.980i)T \)
89 \( 1 + (-0.254 + 0.967i)T \)
97 \( 1 + (-0.198 - 0.980i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.275477647149298819737307336672, −21.88823201845073133315711390036, −21.21994165564852911845120804363, −20.691600633445806988207663080765, −20.01837951231019080740535728882, −19.00994892667291142613249670299, −18.48681233267747598169712429698, −17.604224167963245687852760037651, −16.625856966043955122011609108651, −15.63873463074027834934094195870, −14.47100362747680759774267510779, −13.90894524109981361503945648730, −12.942369604744314988014135525492, −12.09975841097607371720385580430, −11.05946852141377853660566872658, −10.181529676822037314073749296065, −9.51170196597706114284508512916, −8.32671388980167839479649981009, −7.969031024097862371640612839, −6.99863457845263880211370058743, −5.08624674182121122509388256820, −4.171976144924724180049376417827, −3.29703741848738178847783682499, −2.06554743508905538742976636728, −1.4794578197001995106711187574, 0.975263035405622844429093439763, 2.264187150229257168298875396879, 3.4024844670570184151500495161, 4.98579741105549800775344878245, 5.38543352211796894942000915569, 7.002599898804422453077694731122, 7.68291016069017605615349265538, 8.39297515136437029400155335387, 9.0891346234555601452818199677, 10.159341452226788542927970276515, 10.90668941570384772754266123011, 12.3322108542267768991668129190, 13.57545660148464314715584727480, 13.972417043232047638768474959663, 15.04433344200130023337264841762, 15.517103482959602817740162319565, 16.33869605187742595011726623416, 17.6218834005725065538104218893, 18.2826235232772028095115240873, 18.76691467274057120586016493512, 19.989855861823365870614831045642, 20.4996464119028513536163910798, 21.50373986534811438312509696304, 22.51404423943943521732184812556, 23.788599956807085761813392434686

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