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Properties

Label 1-1001-1001.125-r0-0-0
Degree $1$
Conductor $1001$
Sign $-0.834 + 0.551i$
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  + (−0.951 + 0.309i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.951 − 0.309i)5-s + (−0.951 − 0.309i)6-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + 10-s + 12-s + (−0.587 − 0.809i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (−0.587 − 0.809i)18-s + (−0.587 + 0.809i)19-s + (−0.951 + 0.309i)20-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.951 − 0.309i)5-s + (−0.951 − 0.309i)6-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s + 10-s + 12-s + (−0.587 − 0.809i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (−0.587 − 0.809i)18-s + (−0.587 + 0.809i)19-s + (−0.951 + 0.309i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.834 + 0.551i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ -0.834 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1841687781 + 0.6122571320i\)
\(L(\frac12)\) \(\approx\) \(0.1841687781 + 0.6122571320i\)
\(L(1)\) \(\approx\) \(0.6372859551 + 0.2766612754i\)
\(L(1)\) \(\approx\) \(0.6372859551 + 0.2766612754i\)

Euler product

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

−21.09606519318565218321941777961, −20.15160480674696158368542877258, −19.678348233971151212401268823782, −19.06246525043486308337909915672, −18.46791548449731602399848491870, −17.62555009020959764088407833393, −16.79743845673581704045680905779, −15.66096367296959635480145669682, −15.24414176721845583140065005899, −14.34347136876030057538051673891, −13.193123303652751548432695833591, −12.40261664503747987830631728344, −11.75721809757212820892088612066, −10.84784741048397243787293685970, −9.99195564834366867730677532536, −9.006412403774480246287617777370, −8.24776365629692374306054631628, −7.75015052148954566177376395178, −6.89968392020327516990488108834, −6.136618652535986097078250913903, −4.21384251159403165634010185634, −3.498614865833531282354363523968, −2.56424399192231045560946730837, −1.67321857307154520777083209566, −0.34444123628380936449964940922, 1.30689112666946186583135248219, 2.49964164912570563590912559600, 3.47169413405951456075089730134, 4.472144476617522768601778035114, 5.4469336936974567256879152596, 6.71876471184704353090638502171, 7.72172391108946830315113270068, 8.15144393561750131593616629246, 8.93416526568898536104298640647, 9.784902246740280291946239901697, 10.45152136592930036748175046828, 11.45936338056708184108014693692, 12.138137083019935686693315893506, 13.4164920423943077394532589526, 14.4252906454959278471461361147, 15.09123953918619542975184092742, 15.72885152104150690693412191246, 16.48767689529336100038407400409, 16.92606398001336422667146885233, 18.43983023468222953740804154265, 18.74216144718449982612009229106, 19.74126783778908630459465391185, 20.2408073036272355455478742044, 20.77688123101950274333097649063, 21.78090519088603775076449098282

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