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Properties

Label 1-2585-2585.1052-r0-0-0
Degree $1$
Conductor $2585$
Sign $0.887 + 0.460i$
Analytic cond. $12.0046$
Root an. cond. $12.0046$
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  + (−0.609 − 0.792i)2-s + (−0.911 − 0.410i)3-s + (−0.256 + 0.966i)4-s + (0.230 + 0.973i)6-s + (−0.999 + 0.0136i)7-s + (0.922 − 0.385i)8-s + (0.662 + 0.749i)9-s + (0.631 − 0.775i)12-s + (−0.423 + 0.905i)13-s + (0.620 + 0.784i)14-s + (−0.868 − 0.496i)16-s + (−0.163 − 0.986i)17-s + (0.190 − 0.981i)18-s + (−0.531 + 0.847i)19-s + (0.917 + 0.398i)21-s + ⋯
L(s)  = 1  + (−0.609 − 0.792i)2-s + (−0.911 − 0.410i)3-s + (−0.256 + 0.966i)4-s + (0.230 + 0.973i)6-s + (−0.999 + 0.0136i)7-s + (0.922 − 0.385i)8-s + (0.662 + 0.749i)9-s + (0.631 − 0.775i)12-s + (−0.423 + 0.905i)13-s + (0.620 + 0.784i)14-s + (−0.868 − 0.496i)16-s + (−0.163 − 0.986i)17-s + (0.190 − 0.981i)18-s + (−0.531 + 0.847i)19-s + (0.917 + 0.398i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2585 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2585 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2585\)    =    \(5 \cdot 11 \cdot 47\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(12.0046\)
Root analytic conductor: \(12.0046\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2585} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2585,\ (0:\ ),\ 0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3670509867 + 0.08961129993i\)
\(L(\frac12)\) \(\approx\) \(0.3670509867 + 0.08961129993i\)
\(L(1)\) \(\approx\) \(0.4462741890 - 0.1520641979i\)
\(L(1)\) \(\approx\) \(0.4462741890 - 0.1520641979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.609 - 0.792i)T \)
3 \( 1 + (-0.911 - 0.410i)T \)
7 \( 1 + (-0.999 + 0.0136i)T \)
13 \( 1 + (-0.423 + 0.905i)T \)
17 \( 1 + (-0.163 - 0.986i)T \)
19 \( 1 + (-0.531 + 0.847i)T \)
23 \( 1 + (0.942 + 0.334i)T \)
29 \( 1 + (-0.981 - 0.190i)T \)
31 \( 1 + (0.994 - 0.109i)T \)
37 \( 1 + (-0.784 - 0.620i)T \)
41 \( 1 + (0.996 + 0.0818i)T \)
43 \( 1 + (-0.631 - 0.775i)T \)
53 \( 1 + (0.652 - 0.758i)T \)
59 \( 1 + (-0.998 - 0.0546i)T \)
61 \( 1 + (-0.554 + 0.832i)T \)
67 \( 1 + (0.816 - 0.576i)T \)
71 \( 1 + (0.792 + 0.609i)T \)
73 \( 1 + (0.216 + 0.976i)T \)
79 \( 1 + (-0.435 - 0.900i)T \)
83 \( 1 + (0.711 - 0.702i)T \)
89 \( 1 + (0.0682 - 0.997i)T \)
97 \( 1 + (-0.496 - 0.868i)T \)
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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

−19.22670035552343263545374113497, −18.462116916632065126546505106066, −17.69001290731282838580918232417, −17.00354845535233111251706475333, −16.77148644024062940664723876755, −15.75060694491934199875355258438, −15.305088366035123386492889897250, −14.832115351462758487936147656411, −13.54803307768253077733933527696, −12.88541404366549771124033150169, −12.243187537427669665783977390571, −10.92842523489046075972420325344, −10.69471200925690723768649722868, −9.81971917645719379783968168181, −9.28713274514241255942105988459, −8.43234814475248229998010196941, −7.45582813592361728079212890560, −6.62633321977081233165680379586, −6.243435452770730885670769636699, −5.361609769449279958126460160, −4.71480091320322045202342046326, −3.75558746194604850929669960726, −2.64073553854159977902347887387, −1.24349750795350885760938144584, −0.27313195509553689343784339262, 0.7123128845114892539360334849, 1.80516160129107502509119906802, 2.5418760863408826789464545584, 3.59381256782545467964648004438, 4.40883760275441189282898289810, 5.30782920384166557980517794122, 6.34051455845620461363881614958, 7.056734992163784185420625613301, 7.56613535296310915768793053427, 8.73663231809443324105918299568, 9.46306486858891009225238858798, 10.06720254272737495063156951451, 10.82193725509199465029692344771, 11.59353244778498164927781463197, 12.09813775045612075440078029304, 12.806939127585970286805676272075, 13.38273806776355890352600171526, 14.15829836080727193708231619016, 15.51973127250342631379262040947, 16.21139461760166781931798289038, 16.87579277647097398510088879641, 17.208707967913484400493463353266, 18.22361421764146994603266220913, 18.82127438975797512123427103887, 19.19179498014920033597576479817

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