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Properties

Label 1-1001-1001.237-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.651 + 0.758i$
Analytic cond. $4.64862$
Root an. cond. $4.64862$
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  + (0.104 + 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (−0.104 + 0.994i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s − 12-s + (0.669 + 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + (0.669 − 0.743i)19-s + (−0.913 − 0.406i)20-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.978 + 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.809 + 0.587i)5-s + (−0.104 + 0.994i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s − 12-s + (0.669 + 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.104 + 0.994i)17-s + (−0.309 + 0.951i)18-s + (0.669 − 0.743i)19-s + (−0.913 − 0.406i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.651 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9970036372 + 2.168990347i\)
\(L(\frac12)\) \(\approx\) \(0.9970036372 + 2.168990347i\)
\(L(1)\) \(\approx\) \(1.205280749 + 1.084700206i\)
\(L(1)\) \(\approx\) \(1.205280749 + 1.084700206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−21.04328485816625706756444652152, −20.5693058394706397893637625857, −20.08756645122239236633019615448, −19.16016840018488980543753714140, −18.289481124413053477278193890275, −17.926600624963933918386925435207, −16.73389439730088195406824682843, −15.84314314511230615635674323120, −14.559328897080064789238855782715, −14.06103324403606345583374278620, −13.44846762051068306613708826966, −12.59470346054225252336170129784, −12.08580261447371126759715773678, −10.82684783164223697812843134080, −9.86363296613750697180327782893, −9.42743435145899891909197351322, −8.61432813589715088026051517561, −7.84930383186927020905976659623, −6.56394176508759064391362226196, −5.38481643019780359944687837511, −4.57907230558747870018646153052, −3.55394810035529225711197194372, −2.63675348009611888060303549351, −1.8564087598843109729725190632, −0.94105387142941036015367204361, 1.49259356392337756721601690107, 2.68464899548728239776529454193, 3.58981440619180863906006996702, 4.51355021718701375777406639418, 5.59027841703501229197168448050, 6.40614582329282624405080963030, 7.30868548549518342037610257260, 8.02892938055418269296921832876, 8.93443732825532627107676547893, 9.726210840650587445029754875730, 10.21369186597833234985587328252, 11.55395337797681215195724955580, 12.99464325108679753880808457802, 13.45572019148910445512951246498, 14.071563191580896632685347496124, 15.00943614492897109799653064822, 15.29685496053117324221392690261, 16.32095271742173051223706668959, 17.269224404230324502653968062997, 17.89810851453892832649742889307, 18.77116769714531509421110046622, 19.38656818901871223087451156929, 20.47616074765728273426633874217, 21.40906343513865832244729100010, 21.93211863745164286353791303622

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