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Properties

Label 1-1625-1625.6-r1-0-0
Degree $1$
Conductor $1625$
Sign $0.273 + 0.961i$
Analytic cond. $174.630$
Root an. cond. $174.630$
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  + (−0.999 + 0.0209i)2-s + (0.146 + 0.989i)3-s + (0.999 − 0.0418i)4-s + (−0.166 − 0.985i)6-s + (−0.743 + 0.669i)7-s + (−0.998 + 0.0627i)8-s + (−0.957 + 0.289i)9-s + (0.999 − 0.0209i)11-s + (0.187 + 0.982i)12-s + (0.728 − 0.684i)14-s + (0.996 − 0.0836i)16-s + (−0.463 − 0.886i)17-s + (0.951 − 0.309i)18-s + (−0.989 − 0.146i)19-s + (−0.770 − 0.637i)21-s + (−0.999 + 0.0418i)22-s + ⋯
L(s)  = 1  + (−0.999 + 0.0209i)2-s + (0.146 + 0.989i)3-s + (0.999 − 0.0418i)4-s + (−0.166 − 0.985i)6-s + (−0.743 + 0.669i)7-s + (−0.998 + 0.0627i)8-s + (−0.957 + 0.289i)9-s + (0.999 − 0.0209i)11-s + (0.187 + 0.982i)12-s + (0.728 − 0.684i)14-s + (0.996 − 0.0836i)16-s + (−0.463 − 0.886i)17-s + (0.951 − 0.309i)18-s + (−0.989 − 0.146i)19-s + (−0.770 − 0.637i)21-s + (−0.999 + 0.0418i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $0.273 + 0.961i$
Analytic conductor: \(174.630\)
Root analytic conductor: \(174.630\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (1:\ ),\ 0.273 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5818468374 + 0.4393101939i\)
\(L(\frac12)\) \(\approx\) \(0.5818468374 + 0.4393101939i\)
\(L(1)\) \(\approx\) \(0.5541057474 + 0.2029778664i\)
\(L(1)\) \(\approx\) \(0.5541057474 + 0.2029778664i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.999 + 0.0209i)T \)
3 \( 1 + (0.146 + 0.989i)T \)
7 \( 1 + (-0.743 + 0.669i)T \)
11 \( 1 + (0.999 - 0.0209i)T \)
17 \( 1 + (-0.463 - 0.886i)T \)
19 \( 1 + (-0.989 - 0.146i)T \)
23 \( 1 + (-0.604 - 0.796i)T \)
29 \( 1 + (-0.699 - 0.714i)T \)
31 \( 1 + (-0.844 - 0.535i)T \)
37 \( 1 + (0.0836 + 0.996i)T \)
41 \( 1 + (0.921 - 0.387i)T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (0.998 + 0.0627i)T \)
53 \( 1 + (-0.637 + 0.770i)T \)
59 \( 1 + (-0.328 + 0.944i)T \)
61 \( 1 + (0.387 - 0.921i)T \)
67 \( 1 + (-0.963 + 0.268i)T \)
71 \( 1 + (0.553 - 0.832i)T \)
73 \( 1 + (-0.982 - 0.187i)T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (0.368 + 0.929i)T \)
89 \( 1 + (0.328 + 0.944i)T \)
97 \( 1 + (0.963 + 0.268i)T \)
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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

−19.82692372407469383968474922982, −19.39375405577579897229258780097, −18.73253419832550364214888708756, −17.80930503398568893542790191825, −17.2417708380397767757031482707, −16.70151967340644756013707956176, −15.85725438225977433023360260285, −14.771232543645787481607278602666, −14.232368284125352078497039048202, −13.01307639910235118321593837692, −12.65217803144441668737365955633, −11.67731790283841818642661795262, −10.95575721251065596978357940443, −10.163845595775427626143328810133, −9.13975489591113145524434479757, −8.712844982633500626037225004859, −7.65961046702833430079624786924, −7.078630403839698992034929915224, −6.367813799465411331999177003487, −5.7916055719402823627592159569, −3.96390679855628551515560314292, −3.29131475997696729607097559583, −2.03592007416025789142672467594, −1.47376346138437690775258898285, −0.37520308727321419472004189921, 0.44411583483843523795050601960, 2.02690316426789553083224473735, 2.73295982409919613038662203737, 3.68106526820632347853948967357, 4.61365093608602389852338481875, 5.92038981711767013029368396519, 6.31963604556424336027086775866, 7.4037651248227476960353182399, 8.49666338379847131508444265772, 9.066584204756127446772369076128, 9.55412804796280482685863078396, 10.3110714242469610965855886437, 11.19107760396655337002615203986, 11.78207213026463863173506290142, 12.64290150038731809006750912741, 13.836942698214982071488105354576, 14.85534197032361881373112149952, 15.263252116272795342739394111307, 16.09757062325904314693243861554, 16.6634723294502219872936917063, 17.23703090820263712814376073032, 18.24003256253464948637769393458, 18.98140985454326841220627058045, 19.647854841274338273885393907413, 20.316299363384981498464904828412

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