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Properties

Label 1-40e2-1600.829-r0-0-0
Degree $1$
Conductor $1600$
Sign $0.999 + 0.00392i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (−0.0784 + 0.996i)3-s + (−0.707 + 0.707i)7-s + (−0.987 − 0.156i)9-s + (−0.972 − 0.233i)11-s + (−0.233 − 0.972i)13-s + (−0.951 − 0.309i)17-s + (0.760 + 0.649i)19-s + (−0.649 − 0.760i)21-s + (−0.156 − 0.987i)23-s + (0.233 − 0.972i)27-s + (−0.0784 + 0.996i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.852 − 0.522i)37-s + (0.987 − 0.156i)39-s + ⋯
L(s)  = 1  + (−0.0784 + 0.996i)3-s + (−0.707 + 0.707i)7-s + (−0.987 − 0.156i)9-s + (−0.972 − 0.233i)11-s + (−0.233 − 0.972i)13-s + (−0.951 − 0.309i)17-s + (0.760 + 0.649i)19-s + (−0.649 − 0.760i)21-s + (−0.156 − 0.987i)23-s + (0.233 − 0.972i)27-s + (−0.0784 + 0.996i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.852 − 0.522i)37-s + (0.987 − 0.156i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.999 + 0.00392i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ 0.999 + 0.00392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8219270100 + 0.001613851984i\)
\(L(\frac12)\) \(\approx\) \(0.8219270100 + 0.001613851984i\)
\(L(1)\) \(\approx\) \(0.7607870081 + 0.2080901182i\)
\(L(1)\) \(\approx\) \(0.7607870081 + 0.2080901182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.0784 + 0.996i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.972 - 0.233i)T \)
13 \( 1 + (-0.233 - 0.972i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.760 + 0.649i)T \)
23 \( 1 + (-0.156 - 0.987i)T \)
29 \( 1 + (-0.0784 + 0.996i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.852 - 0.522i)T \)
41 \( 1 + (0.987 + 0.156i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.649 - 0.760i)T \)
59 \( 1 + (-0.852 - 0.522i)T \)
61 \( 1 + (0.522 + 0.852i)T \)
67 \( 1 + (0.649 - 0.760i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.156 + 0.987i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (-0.760 - 0.649i)T \)
89 \( 1 + (0.987 - 0.156i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−20.28715902110048088006621187676, −19.60819048679168672019484501634, −19.04869073232984730997454322892, −18.3043118364247555030093676368, −17.42625930931676068414027163115, −17.034267617208789923043490565499, −15.9386728370927251313889608912, −15.40453657810916989597985646412, −14.11270583987836300954164878620, −13.61105569879804490079967984118, −13.053578409698113021101928715579, −12.28827359522443541301233461763, −11.37280605573354885633483166092, −10.783201731237990045613928470398, −9.63898121298099131919075819513, −9.062327346099261530443627411851, −7.77836209214639149899246462629, −7.41738592368796136163901419270, −6.56786062480300676701191169175, −5.85991883338791724645005000083, −4.79288391232518758211661211625, −3.81830819420653572999175163869, −2.70214383501144138909215020418, −2.00788163476017096986843689110, −0.77817520429226015825591634566, 0.410523548257906117026703722084, 2.31825952618584527872754945096, 2.99408528482272505794426716988, 3.74166157089545376880522290288, 4.97867445117799643228223684764, 5.43476221671939715696255592843, 6.25171146428191655337714538250, 7.36718751573336302162199514186, 8.42297882455428187248870117379, 8.99339109451632180607770021151, 9.85952137651120456573261779619, 10.54500146754555739005398038237, 11.111414274209803852833978313751, 12.33042541716293694167586854202, 12.7068627403706693240205489840, 13.847879146384402852556551504185, 14.58665826730857354967636124687, 15.523392454212264110236441298964, 15.903364460811601176721912524941, 16.42758860731696852291334153894, 17.5735325156017151453663293788, 18.15350807446603263271368040359, 18.97202305514609238157975412146, 20.0072021129667982644430578639, 20.40657668230192272403923882946

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