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Properties

Label 1-1925-1925.1161-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.986 + 0.165i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)16-s + 17-s + (−0.309 + 0.951i)18-s + (−0.809 − 0.587i)19-s + (0.309 − 0.951i)23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)16-s + 17-s + (−0.309 + 0.951i)18-s + (−0.809 − 0.587i)19-s + (0.309 − 0.951i)23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.986 + 0.165i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.986 + 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008969551930 - 0.1074869510i\)
\(L(\frac12)\) \(\approx\) \(0.008969551930 - 0.1074869510i\)
\(L(1)\) \(\approx\) \(0.6016865514 - 0.1160020729i\)
\(L(1)\) \(\approx\) \(0.6016865514 - 0.1160020729i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 - T \)
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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

−20.00934252870937782749298077367, −19.46749460387335558912009100040, −18.67724209825388147618020810212, −18.344108434326678628037068749530, −17.32082226271158267371610971783, −16.848360713295202652275010683444, −16.38578994785014718538881964018, −15.158199943922890646540032144123, −14.62592560761654945960248534696, −13.85007930711693245721374277169, −13.21302424957182937175572946730, −12.35093268807146980451624807290, −11.69826038400180032176998233620, −10.61588509057996249153099398369, −9.84986176537155183622820576464, −8.96934290801573011427813308964, −8.12226817559014877581622367563, −7.51485356305040536385370099212, −6.84773285453462672827139064106, −6.0823036741033820535643867004, −5.37270418637268870172536101689, −4.577988843581139680215597207701, −3.38083603056000882026957075320, −2.049433637867533057712188122120, −1.24006466072340376874366053300, 0.049403336685336123398954420164, 1.22125564375899886443488751977, 2.67910958161666599546008892208, 3.05782210753914809384278125406, 4.2240525817425006759512417045, 4.761920765803438382127275675519, 5.56418728712853886829666080746, 6.690888429581493852940800518613, 7.88045822330785614860393616352, 8.580109330681430332904374373733, 9.36697729588405592451874845942, 10.174205961244015473457135622573, 10.48905266350022328019515531169, 11.38830226222050652222061566654, 12.12203806540338389424637936303, 12.70277398307716390941672874969, 13.68594885327748392415540052600, 14.64738456776430382935052328384, 15.08393132121320701718259474767, 16.30369288390429355299098693910, 16.78711645203398173775936566071, 17.50786920461763158339922830225, 18.141282698638286132944596515543, 19.091621708958448344553488033816, 19.80668296602904574384299777660

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