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Properties

Label 1-247-247.220-r0-0-0
Degree $1$
Conductor $247$
Sign $-0.977 - 0.211i$
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  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s − 7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s − 11-s + 12-s + (−0.5 − 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s − 7-s − 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s − 11-s + 12-s + (−0.5 − 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(1.14706\)
Root analytic conductor: \(1.14706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (0:\ ),\ -0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04682380069 + 0.4383009025i\)
\(L(\frac12)\) \(\approx\) \(-0.04682380069 + 0.4383009025i\)
\(L(1)\) \(\approx\) \(0.6425504555 + 0.3906612508i\)
\(L(1)\) \(\approx\) \(0.6425504555 + 0.3906612508i\)

Euler product

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

−25.82888254560785400785283800114, −24.324226094808587822590490257885, −23.5437504233425087728013516626, −22.546452408935643239883488925210, −21.89150688294252558065712652537, −20.97276595033677613644437207655, −20.402447434969973485734231809686, −19.38438483079968992594940974562, −18.16869411298501239121539524881, −17.149295230477454006230049640746, −16.08920350319694666069134107621, −15.35687506125661494102430589506, −14.04689101124874703815905589120, −12.87434405167841434891484484224, −12.49021021766108240797414496477, −11.09454602240154053486258787490, −10.21162742416548414303832637581, −9.536244889093945269188560266111, −8.55872532125706732374432012548, −6.28690561346365758098530539133, −5.54254150902563278898452947034, −4.54231150768526006368983704560, −3.54014639396009073295393810137, −2.17401368564998712497231385557, −0.25714117033175265431768053573, 2.36462281215049830784448840435, 3.39307432028497268374236538478, 5.24536772304179094012070883536, 5.965658452786493491515385958137, 6.9972683308426322153046409851, 7.4763492314152410455591165435, 8.99592411228139390404478812667, 10.28818737053921171618103699734, 11.51190450183049137611453772913, 12.69898818077017294572377707010, 13.40110819880293591194626332087, 14.073895767030138215980329047076, 15.39608397768088241966414578539, 16.23721689969631055162287055318, 17.24499810254539262996267736283, 18.25834726687344119174542499935, 18.60155216634728334049842602376, 20.02161981228176066957670985822, 21.56412100126209260017649998927, 22.28967713386860780956729508628, 22.998525722944527690106615388620, 23.70083375388064036677292913841, 24.73743815564581221700211577170, 25.68161996172918607221358484673, 26.04220417192504429433656062291

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