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Properties

Label 1-247-247.43-r0-0-0
Degree $1$
Conductor $247$
Sign $0.122 + 0.992i$
Analytic cond. $1.14706$
Root an. cond. $1.14706$
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  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)14-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s − 18-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.173 − 0.984i)10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)14-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $0.122 + 0.992i$
Analytic conductor: \(1.14706\)
Root analytic conductor: \(1.14706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (0:\ ),\ 0.122 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.676487328 + 1.482389288i\)
\(L(\frac12)\) \(\approx\) \(1.676487328 + 1.482389288i\)
\(L(1)\) \(\approx\) \(1.643640653 + 0.8740918415i\)
\(L(1)\) \(\approx\) \(1.643640653 + 0.8740918415i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−25.57210836180795137626770711171, −24.82179985107018327086778286889, −23.78228555851954997672805585211, −23.08677228729481894990563927094, −22.545257066077828511962915253435, −21.29655448312792438672168728853, −19.9963430698426612432564362513, −19.83850749559835854070776012249, −18.45542001265426008018042802509, −17.80658843673831796899640834230, −16.41670639102687489003782317143, −15.007775840524646400724975156239, −14.25430136718388902255885247223, −13.786998980923332903682664475266, −12.52789307310152613168690292971, −11.701587602405390045540985249459, −10.86209815113183736371639996875, −9.75662438134534274508249043293, −7.84764533962447189446532658648, −7.03606740047424669305494193843, −6.38920304307745870003824957771, −4.85282912716529094074775156647, −3.6117314542549278587907814178, −2.51626409500325828253563744464, −1.36202781964140404840304973479, 2.040091704230972926929705378695, 3.52842450103621286270089722705, 4.37281463047536497000741913, 5.412159171556206293844351145341, 6.08675142031729202047901511594, 8.195812523871087487407627244190, 8.4820534090412017887344778770, 9.888906891547692609947858738293, 11.431074993987573958000848673, 11.86670778354038897021759679516, 13.16512834093592450626023327573, 14.15243896659147223219265758173, 15.12781646481748327932460210621, 15.77074949130148840460610934820, 16.69345343597937832374493588830, 17.36943036959957442484211384994, 19.223500220010343418723730583122, 20.1959645904002997615926951758, 21.08051916611721811485246179471, 21.64301001025876682073696867117, 22.38258143792465428076318348978, 23.66973257279472402355438929589, 24.40864737956008729082328442748, 25.18139916311832558249147367095, 26.127385309203690166731246660655

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