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Properties

Label 1-1625-1625.614-r0-0-0
Degree $1$
Conductor $1625$
Sign $-0.783 + 0.621i$
Analytic cond. $7.54646$
Root an. cond. $7.54646$
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  + (−0.832 − 0.553i)2-s + (0.570 + 0.821i)3-s + (0.387 + 0.921i)4-s + (−0.0209 − 0.999i)6-s + (0.104 − 0.994i)7-s + (0.187 − 0.982i)8-s + (−0.348 + 0.937i)9-s + (0.832 + 0.553i)11-s + (−0.535 + 0.844i)12-s + (−0.637 + 0.770i)14-s + (−0.699 + 0.714i)16-s + (−0.604 − 0.796i)17-s + (0.809 − 0.587i)18-s + (−0.570 + 0.821i)19-s + (0.876 − 0.481i)21-s + (−0.387 − 0.921i)22-s + ⋯
L(s)  = 1  + (−0.832 − 0.553i)2-s + (0.570 + 0.821i)3-s + (0.387 + 0.921i)4-s + (−0.0209 − 0.999i)6-s + (0.104 − 0.994i)7-s + (0.187 − 0.982i)8-s + (−0.348 + 0.937i)9-s + (0.832 + 0.553i)11-s + (−0.535 + 0.844i)12-s + (−0.637 + 0.770i)14-s + (−0.699 + 0.714i)16-s + (−0.604 − 0.796i)17-s + (0.809 − 0.587i)18-s + (−0.570 + 0.821i)19-s + (0.876 − 0.481i)21-s + (−0.387 − 0.921i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1625\)    =    \(5^{3} \cdot 13\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(7.54646\)
Root analytic conductor: \(7.54646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1625} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1625,\ (0:\ ),\ -0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1677449823 + 0.4817977623i\)
\(L(\frac12)\) \(\approx\) \(0.1677449823 + 0.4817977623i\)
\(L(1)\) \(\approx\) \(0.7183801493 + 0.09875895655i\)
\(L(1)\) \(\approx\) \(0.7183801493 + 0.09875895655i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.832 - 0.553i)T \)
3 \( 1 + (0.570 + 0.821i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (0.832 + 0.553i)T \)
17 \( 1 + (-0.604 - 0.796i)T \)
19 \( 1 + (-0.570 + 0.821i)T \)
23 \( 1 + (-0.783 + 0.621i)T \)
29 \( 1 + (-0.957 + 0.289i)T \)
31 \( 1 + (-0.992 + 0.125i)T \)
37 \( 1 + (0.699 - 0.714i)T \)
41 \( 1 + (0.146 - 0.989i)T \)
43 \( 1 + (-0.669 + 0.743i)T \)
47 \( 1 + (0.187 + 0.982i)T \)
53 \( 1 + (-0.876 + 0.481i)T \)
59 \( 1 + (-0.999 - 0.0418i)T \)
61 \( 1 + (0.146 + 0.989i)T \)
67 \( 1 + (-0.228 - 0.973i)T \)
71 \( 1 + (-0.756 + 0.653i)T \)
73 \( 1 + (-0.535 - 0.844i)T \)
79 \( 1 + (-0.425 + 0.904i)T \)
83 \( 1 + (0.425 + 0.904i)T \)
89 \( 1 + (-0.999 + 0.0418i)T \)
97 \( 1 + (-0.228 + 0.973i)T \)
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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

−19.90323005487436109229555600020, −19.219279107603056945736322466016, −18.632164717442065041962050229091, −18.066250188070550215741883889310, −17.26949708590670568801287492813, −16.606465564449110850948717917272, −15.53956013194718434198952641283, −14.91090755278033177742396282235, −14.44899245131066739773061263368, −13.44297224393788021300492048339, −12.68153197516634917894748233648, −11.639090696809534351315166395125, −11.18736813971701304838400279064, −9.93590618485749992473144778763, −9.03202522511454624395375800029, −8.66271144453836050645048778857, −8.004982948588959539088940204231, −7.01345680706745957705456418841, −6.24890919585257449442718757328, −5.835638497959738966067959782469, −4.490281085165527194789189446138, −3.20512241261660292980598665767, −2.12382789608529402375489712216, −1.63750128671862210599254659751, −0.21208041585698709133715890567, 1.44297707327313196772468763467, 2.19143565776173142962593766504, 3.38041261216694234315795129548, 3.98913356554658460690024995919, 4.625766057660733784728742456629, 6.07794513573716365924937638696, 7.29967293822382730904953129091, 7.65880049145687029696244203278, 8.730589292901042767971206463948, 9.42323536562492844203221534406, 9.9132780444696116180345384592, 10.83369993310588397315639885429, 11.25435596073125683055440584379, 12.31042004138902535730016572304, 13.20205258206496319161264649755, 14.055695730651403995331171402788, 14.726091928169830483064675339829, 15.71725754500256620441403822555, 16.44877615729363067890031346576, 16.96035604270592784266741522980, 17.72681914851960728061588153413, 18.56095621961199383569619143374, 19.672671714353148196729836909661, 19.85684095236388875024946416468, 20.615035166602514688530986303256

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