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Properties

Label 1-1001-1001.191-r0-0-0
Degree $1$
Conductor $1001$
Sign $0.0283 - 0.999i$
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  + (−0.978 − 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 + 0.406i)4-s + (−0.978 + 0.207i)5-s + (0.669 + 0.743i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s + (−0.5 − 0.866i)12-s + (0.913 + 0.406i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + (−0.809 − 0.587i)19-s + (−0.978 − 0.207i)20-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.809 − 0.587i)3-s + (0.913 + 0.406i)4-s + (−0.978 + 0.207i)5-s + (0.669 + 0.743i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s + (−0.5 − 0.866i)12-s + (0.913 + 0.406i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + (−0.809 − 0.587i)19-s + (−0.978 − 0.207i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0283 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0283 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.0283 - 0.999i$
Analytic conductor: \(4.64862\)
Root analytic conductor: \(4.64862\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1001,\ (0:\ ),\ 0.0283 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2164706703 - 0.2104133332i\)
\(L(\frac12)\) \(\approx\) \(0.2164706703 - 0.2104133332i\)
\(L(1)\) \(\approx\) \(0.3906382004 - 0.08274010834i\)
\(L(1)\) \(\approx\) \(0.3906382004 - 0.08274010834i\)

Euler product

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

−21.87064173812635756898471167136, −20.78263268178985360116178752445, −20.29797177548844179389077197217, −19.41902641142964336745950999843, −18.59102485039049024446193675361, −17.91468782435017029505884180713, −16.950167654751531289904080957372, −16.52300370710177096174397673160, −15.61020209731454240389559220122, −15.29508223173547206361273395936, −14.28072406582419593108096602400, −12.679688125933262331560337711350, −12.00482209621419956795360407674, −11.26912863738461762141033022461, −10.5932361995345888179815030204, −9.87631669056234038285367275689, −8.75165615987308369773017053459, −8.29430459359935025585925895332, −7.033874605455411503453959577094, −6.50225546897220194064100195020, −5.39582602762556924592545912233, −4.44642909000474314436251992825, −3.52314963350871516629433344058, −2.11966798136534740876967298623, −0.68983761981291137195092560330, 0.322728320896129905883498925266, 1.58516769180478702247827627778, 2.58810945750661105120363642991, 3.80567966300896360795584572411, 4.87731781277809873534784880265, 6.25105664719927833040917664981, 6.80554919958653022110384162133, 7.67011185943700129757090938945, 8.29389088778165571170852265495, 9.29547064204031849520621161174, 10.45931568861872198825954917801, 11.14967993953159348666588136169, 11.54629339837747447369804655859, 12.53608779436505901624658210971, 13.07829031976983806126179773548, 14.51974480946126314556887794615, 15.58010704660264170372524604462, 16.05431880037841943025600146503, 16.89953159606914071039111208710, 17.79288140841875170297480895694, 18.151540651233024216455703783167, 19.29870046097650253352035044393, 19.48511189840655181955518403065, 20.33509968275260686480340677355, 21.61469811560146813408347023835

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