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Properties

Label 1-425-425.413-r0-0-0
Degree $1$
Conductor $425$
Sign $-0.256 + 0.966i$
Analytic cond. $1.97369$
Root an. cond. $1.97369$
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  + (−0.453 + 0.891i)2-s + (0.972 − 0.233i)3-s + (−0.587 − 0.809i)4-s + (−0.233 + 0.972i)6-s + (0.382 + 0.923i)7-s + (0.987 − 0.156i)8-s + (0.891 − 0.453i)9-s + (−0.760 + 0.649i)11-s + (−0.760 − 0.649i)12-s + (−0.309 + 0.951i)13-s + (−0.996 − 0.0784i)14-s + (−0.309 + 0.951i)16-s + i·18-s + (−0.987 + 0.156i)19-s + (0.587 + 0.809i)21-s + (−0.233 − 0.972i)22-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s + (0.972 − 0.233i)3-s + (−0.587 − 0.809i)4-s + (−0.233 + 0.972i)6-s + (0.382 + 0.923i)7-s + (0.987 − 0.156i)8-s + (0.891 − 0.453i)9-s + (−0.760 + 0.649i)11-s + (−0.760 − 0.649i)12-s + (−0.309 + 0.951i)13-s + (−0.996 − 0.0784i)14-s + (−0.309 + 0.951i)16-s + i·18-s + (−0.987 + 0.156i)19-s + (0.587 + 0.809i)21-s + (−0.233 − 0.972i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.256 + 0.966i$
Analytic conductor: \(1.97369\)
Root analytic conductor: \(1.97369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 425,\ (0:\ ),\ -0.256 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8055221915 + 1.047365030i\)
\(L(\frac12)\) \(\approx\) \(0.8055221915 + 1.047365030i\)
\(L(1)\) \(\approx\) \(0.9542728514 + 0.5493826576i\)
\(L(1)\) \(\approx\) \(0.9542728514 + 0.5493826576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.453 + 0.891i)T \)
3 \( 1 + (0.972 - 0.233i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.760 + 0.649i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.987 + 0.156i)T \)
23 \( 1 + (0.649 + 0.760i)T \)
29 \( 1 + (0.972 - 0.233i)T \)
31 \( 1 + (-0.852 + 0.522i)T \)
37 \( 1 + (0.649 - 0.760i)T \)
41 \( 1 + (0.0784 + 0.996i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (0.649 + 0.760i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.233 - 0.972i)T \)
73 \( 1 + (0.996 + 0.0784i)T \)
79 \( 1 + (0.852 + 0.522i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.233 + 0.972i)T \)
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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

−23.92812939490789671810321409455, −22.93749324308537931934748623985, −21.82096071558921616175968173154, −21.08829594014986171847086186555, −20.43126762464343630775287161847, −19.76094975558097211499893998452, −18.94402284735060310539209045345, −18.09122236718819526351367828731, −17.08732695252356875637441054902, −16.21951204699112500340380123154, −15.01483029513319216294312655226, −14.07942597259083969798447352796, −13.19995141552786568410308737298, −12.6557327001665011462345439610, −11.03695113155327120384232078474, −10.5403624069814107721302129924, −9.72236531306616599261949348454, −8.47609623879293307278511707696, −8.05828814089991155957755141019, −7.02857690070083429172071295771, −5.04901508645853579124225038529, −4.117593045074773252740558006337, −3.11353098338001698102007280587, −2.26557805380003056634471814383, −0.82223612026439149575781079312, 1.62473656498994391292572002891, 2.52567270497039074986117842332, 4.21744157635392551193673214064, 5.14649312254500566596676840844, 6.405381731073479540304932066367, 7.35411207054284525917605891450, 8.151715572497016181499869011467, 9.01317891561645100584907321245, 9.63414718259518950922466701146, 10.81666220626734896018811848976, 12.284273813959363110277757800881, 13.16818290013008384022643722979, 14.20389050013631112911510647066, 14.92059091792120744559614152093, 15.48373567016481738284539401451, 16.45781062423560797411519114724, 17.70346569308760887037266160110, 18.3106147035135586466369217317, 19.1452088517546736374331447231, 19.76753368658680480647567636512, 21.10266780600517716283191268236, 21.63548299871880833652842613391, 23.11462584572290968979604216643, 23.817899409990678903545113751299, 24.60706451959664652695740497828

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