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Properties

Label 1-4025-4025.17-r1-0-0
Degree $1$
Conductor $4025$
Sign $0.960 + 0.277i$
Analytic cond. $432.546$
Root an. cond. $432.546$
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  + (−0.730 − 0.683i)2-s + (0.353 − 0.935i)3-s + (0.0665 + 0.997i)4-s + (−0.897 + 0.441i)6-s + (0.633 − 0.774i)8-s + (−0.749 − 0.662i)9-s + (0.905 − 0.424i)11-s + (0.956 + 0.290i)12-s + (−0.336 + 0.941i)13-s + (−0.991 + 0.132i)16-s + (0.556 − 0.830i)17-s + (0.0950 + 0.995i)18-s + (−0.345 + 0.938i)19-s + (−0.951 − 0.309i)22-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.730 − 0.683i)2-s + (0.353 − 0.935i)3-s + (0.0665 + 0.997i)4-s + (−0.897 + 0.441i)6-s + (0.633 − 0.774i)8-s + (−0.749 − 0.662i)9-s + (0.905 − 0.424i)11-s + (0.956 + 0.290i)12-s + (−0.336 + 0.941i)13-s + (−0.991 + 0.132i)16-s + (0.556 − 0.830i)17-s + (0.0950 + 0.995i)18-s + (−0.345 + 0.938i)19-s + (−0.951 − 0.309i)22-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(432.546\)
Root analytic conductor: \(432.546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (1:\ ),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104764154 + 0.1562595935i\)
\(L(\frac12)\) \(\approx\) \(1.104764154 + 0.1562595935i\)
\(L(1)\) \(\approx\) \(0.7353197354 - 0.3832749132i\)
\(L(1)\) \(\approx\) \(0.7353197354 - 0.3832749132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.730 - 0.683i)T \)
3 \( 1 + (0.353 - 0.935i)T \)
11 \( 1 + (0.905 - 0.424i)T \)
13 \( 1 + (-0.336 + 0.941i)T \)
17 \( 1 + (0.556 - 0.830i)T \)
19 \( 1 + (-0.345 + 0.938i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.161 + 0.986i)T \)
37 \( 1 + (0.662 - 0.749i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (-0.0760 + 0.997i)T \)
59 \( 1 + (-0.179 - 0.983i)T \)
61 \( 1 + (-0.625 - 0.780i)T \)
67 \( 1 + (-0.0190 + 0.999i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (-0.846 - 0.532i)T \)
79 \( 1 + (-0.851 - 0.524i)T \)
83 \( 1 + (0.676 + 0.736i)T \)
89 \( 1 + (-0.449 + 0.893i)T \)
97 \( 1 + (0.676 - 0.736i)T \)
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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

−18.059303716345576072473947630290, −17.34065544046028273051684891817, −16.99037684360122535921976520286, −16.286530128959981219913796434816, −15.43033609358877706051821624796, −15.04505562168344246464499689357, −14.57129920352295645201882662730, −13.77507486280100188482658370878, −12.95713005695222511939585763371, −11.8716933807706908716663534066, −11.14370998866255977954703044856, −10.3322652215875822595937857107, −9.95864377483824092877708304066, −9.1934595384800805738914471261, −8.60940300635727788226853485595, −7.91118223806557496772329166923, −7.23961126427285571758898991242, −6.20928668162750225265320938918, −5.70744645491415627387480764200, −4.695593007005216031665434413930, −4.25994222530241434306034263214, −3.10888068150296905520887796804, −2.30768722826368962721029539969, −1.24132209364126534729449572757, −0.23999523405087065236467069759, 0.85244247144796671170381232952, 1.4207711521091597986569069783, 2.2314201081198990889169134591, 3.01397203995142973977063319547, 3.73381003454675470344863977306, 4.5829798405740075964591307108, 5.881621565730145767696145690924, 6.643585315134038426227889572552, 7.26096017200887740107130853800, 7.92457073540893895764152422652, 8.74557433033604996094330964657, 9.17341478231535145165588115955, 9.9036053160083502331407465983, 10.846380162837466863189702452510, 11.61763348514804741135785945193, 12.11846321343250110634063605029, 12.561393610198888886683133765102, 13.51910921492381586352065883952, 14.21373687745409350158389202091, 14.53723854675255082020009907017, 15.9266924411709492639643688556, 16.52965080687636796189634388028, 17.134661139249400215841411750347, 17.82091026119335859150943012360, 18.51277224415123449552238124914

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