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Properties

Label 1-516-516.287-r1-0-0
Degree $1$
Conductor $516$
Sign $-0.994 + 0.107i$
Analytic cond. $55.4519$
Root an. cond. $55.4519$
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  + (0.826 − 0.563i)5-s + (−0.5 − 0.866i)7-s + (−0.222 + 0.974i)11-s + (0.0747 + 0.997i)13-s + (−0.826 − 0.563i)17-s + (0.955 + 0.294i)19-s + (−0.733 − 0.680i)23-s + (0.365 − 0.930i)25-s + (−0.988 − 0.149i)29-s + (−0.365 − 0.930i)31-s + (−0.900 − 0.433i)35-s + (0.5 − 0.866i)37-s + (−0.623 + 0.781i)41-s + (−0.222 − 0.974i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)5-s + (−0.5 − 0.866i)7-s + (−0.222 + 0.974i)11-s + (0.0747 + 0.997i)13-s + (−0.826 − 0.563i)17-s + (0.955 + 0.294i)19-s + (−0.733 − 0.680i)23-s + (0.365 − 0.930i)25-s + (−0.988 − 0.149i)29-s + (−0.365 − 0.930i)31-s + (−0.900 − 0.433i)35-s + (0.5 − 0.866i)37-s + (−0.623 + 0.781i)41-s + (−0.222 − 0.974i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 516 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 516 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(516\)    =    \(2^{2} \cdot 3 \cdot 43\)
Sign: $-0.994 + 0.107i$
Analytic conductor: \(55.4519\)
Root analytic conductor: \(55.4519\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{516} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 516,\ (1:\ ),\ -0.994 + 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01853267442 - 0.3443879374i\)
\(L(\frac12)\) \(\approx\) \(0.01853267442 - 0.3443879374i\)
\(L(1)\) \(\approx\) \(0.9130030196 - 0.1599979380i\)
\(L(1)\) \(\approx\) \(0.9130030196 - 0.1599979380i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
43 \( 1 \)
good5 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (0.955 + 0.294i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (-0.365 - 0.930i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (-0.955 - 0.294i)T \)
71 \( 1 + (0.733 - 0.680i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.988 + 0.149i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−23.9445527828479984107148810567, −22.62301456414368827279723557784, −22.047295422805650757472604616528, −21.57577368106396293534380715550, −20.43184743701359467513762937193, −19.50660560467369980716085235853, −18.54952881365892761147669597686, −18.01748012926600255333485956686, −17.13332955490248149830726094511, −15.93571596503869855366470815306, −15.37055953853603103400451732740, −14.31148066331892512543796281295, −13.409856652582580495313749019939, −12.78865175775437908177028060214, −11.54282122284231863305439621059, −10.72345534039995533479454547657, −9.786870616161111625298481044643, −8.97155611207683082540187184839, −7.973367856153815176224972657139, −6.716823562964661673092025186126, −5.83853337032826523763436233507, −5.28722263453324379856018223910, −3.45664768047988188799581009632, −2.80629378733767060175444206561, −1.60892765892699734712476602824, 0.080029617278522587395448034170, 1.48944543957185923981048930487, 2.45236334050281097253655044715, 4.0042140449892216996315392799, 4.73906062291158002777143379250, 5.93887450354542252641439300604, 6.8757101438276203933922899196, 7.73148814951922533043058772018, 9.17971579205003379692308887692, 9.627711405600748434228084064083, 10.53865537865324033830803215718, 11.70328323499120924993400916359, 12.6919008285986917812243820596, 13.50614900826763374435264029571, 14.0702561199702129682519406434, 15.23847669140679101813911936077, 16.48505293750723366827646964259, 16.70009244363398362948018087070, 17.90691587615371887057928229033, 18.45276271105641179222649795908, 19.90449327761454321217457128096, 20.30243646746376562177152879626, 21.10732207863206061459805967822, 22.18198751389547646230499812478, 22.812860450040993410992443049834

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