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Properties

Label 1-248-248.27-r0-0-0
Degree $1$
Conductor $248$
Sign $-0.629 + 0.777i$
Analytic cond. $1.15170$
Root an. cond. $1.15170$
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  + (0.809 + 0.587i)3-s − 5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.309 + 0.951i)23-s + 25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.809 + 0.587i)33-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s − 5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)17-s + (−0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.309 + 0.951i)23-s + 25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.809 + 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(248\)    =    \(2^{3} \cdot 31\)
Sign: $-0.629 + 0.777i$
Analytic conductor: \(1.15170\)
Root analytic conductor: \(1.15170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{248} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 248,\ (0:\ ),\ -0.629 + 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4093964412 + 0.8585322073i\)
\(L(\frac12)\) \(\approx\) \(0.4093964412 + 0.8585322073i\)
\(L(1)\) \(\approx\) \(0.8624739402 + 0.4528403573i\)
\(L(1)\) \(\approx\) \(0.8624739402 + 0.4528403573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
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

−26.04995387075962222424843636638, −24.570286591724877626528889184757, −23.93167666554661806902605427785, −23.3639568316027723621843783602, −22.06779788955820013921224157772, −20.90073914806489118365659658822, −19.94793902579910438513139589300, −19.29106240097091493695737401145, −18.74855045104738528198930226720, −17.25254202650715798767426402529, −16.39718737282004853665539732516, −15.2024676937467801012176453236, −14.44925219799341234805403153097, −13.33709426951664081265116372037, −12.648135605279611765914897066395, −11.42427586885467971634141702663, −10.41883887891848130432386128265, −9.01103289532365705372734239136, −8.14551117681864305553847261469, −7.25695239371741173225424635385, −6.39392466066916005665166778925, −4.40996012020708804120165016095, −3.60762615966723168805769927364, −2.36955792329596592976485813180, −0.58761926714768463010853015234, 2.24671526727487123775959327198, 3.18462958509139891227185482925, 4.404269582959937148484565024784, 5.346622000288774706024729885342, 7.17603948534360448819781081217, 7.94635777828900066477016438365, 9.04827704088120454656643568821, 9.84621571353020262481912974972, 11.05280269103555449308564541592, 12.24274497666557309480583727352, 12.99650063657354514142733923314, 14.49986662149885223488967084178, 15.27303735091602161841444749352, 15.68325506058263958045422876280, 16.850426095845115314368342652516, 18.30362794805539049538654179745, 19.13734841695209806099671416340, 20.00008191565974610196690638386, 20.65898114785062761557996325144, 21.86956300854711694831673778043, 22.570696456067246570896735367275, 23.598267410679164403087515744560, 24.9082347450266871053757696122, 25.37615610953864288454456559487, 26.43463862506486910083372485941

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