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Properties

Label 1-643-643.6-r0-0-0
Degree $1$
Conductor $643$
Sign $0.871 - 0.490i$
Analytic cond. $2.98607$
Root an. cond. $2.98607$
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  + (−0.465 − 0.884i)2-s + (0.0440 − 0.999i)3-s + (−0.566 + 0.824i)4-s + (0.957 + 0.289i)5-s + (−0.904 + 0.426i)6-s + (0.916 + 0.399i)7-s + (0.993 + 0.117i)8-s + (−0.996 − 0.0879i)9-s + (−0.189 − 0.981i)10-s + (0.386 + 0.922i)11-s + (0.798 + 0.601i)12-s + (−0.303 + 0.952i)13-s + (−0.0733 − 0.997i)14-s + (0.331 − 0.943i)15-s + (−0.358 − 0.933i)16-s + (−0.516 + 0.856i)17-s + ⋯
L(s)  = 1  + (−0.465 − 0.884i)2-s + (0.0440 − 0.999i)3-s + (−0.566 + 0.824i)4-s + (0.957 + 0.289i)5-s + (−0.904 + 0.426i)6-s + (0.916 + 0.399i)7-s + (0.993 + 0.117i)8-s + (−0.996 − 0.0879i)9-s + (−0.189 − 0.981i)10-s + (0.386 + 0.922i)11-s + (0.798 + 0.601i)12-s + (−0.303 + 0.952i)13-s + (−0.0733 − 0.997i)14-s + (0.331 − 0.943i)15-s + (−0.358 − 0.933i)16-s + (−0.516 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(2.98607\)
Root analytic conductor: \(2.98607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (0:\ ),\ 0.871 - 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245445939 - 0.3262744979i\)
\(L(\frac12)\) \(\approx\) \(1.245445939 - 0.3262744979i\)
\(L(1)\) \(\approx\) \(0.9437825528 - 0.3755310361i\)
\(L(1)\) \(\approx\) \(0.9437825528 - 0.3755310361i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.465 - 0.884i)T \)
3 \( 1 + (0.0440 - 0.999i)T \)
5 \( 1 + (0.957 + 0.289i)T \)
7 \( 1 + (0.916 + 0.399i)T \)
11 \( 1 + (0.386 + 0.922i)T \)
13 \( 1 + (-0.303 + 0.952i)T \)
17 \( 1 + (-0.516 + 0.856i)T \)
19 \( 1 + (-0.0146 - 0.999i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (0.386 + 0.922i)T \)
31 \( 1 + (0.331 + 0.943i)T \)
37 \( 1 + (-0.815 + 0.578i)T \)
41 \( 1 + (-0.780 - 0.625i)T \)
43 \( 1 + (0.680 - 0.732i)T \)
47 \( 1 + (0.916 + 0.399i)T \)
53 \( 1 + (0.972 - 0.232i)T \)
59 \( 1 + (0.0440 + 0.999i)T \)
61 \( 1 + (0.160 + 0.986i)T \)
67 \( 1 + (-0.948 - 0.317i)T \)
71 \( 1 + (0.491 - 0.870i)T \)
73 \( 1 + (0.160 + 0.986i)T \)
79 \( 1 + (0.491 + 0.870i)T \)
83 \( 1 + (0.331 - 0.943i)T \)
89 \( 1 + (-0.465 + 0.884i)T \)
97 \( 1 + (0.275 - 0.961i)T \)
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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

−22.87634820727770650553360469151, −22.21993111654669344444247716352, −21.33784045996540581904945768103, −20.52336531354735047087398574629, −19.82769534240452472997290752518, −18.560767974262845162213162078148, −17.61543625251502677670087461582, −17.193622928820793271177745167211, −16.410694108552065725904139239127, −15.61738338514044923289209709799, −14.67185907965578645586210899504, −13.99035623748711922106263713739, −13.47506962267685455935582694745, −11.764104678598458713104312264162, −10.72322385040634319591882877794, −10.06505717191945087734564844818, −9.315176815924299975040206483676, −8.41316496221110249764825240026, −7.75254719348938081828349855890, −6.2381610890272656792583544243, −5.56409655582286690436162449965, −4.83834260815709513859105868446, −3.82006987465425241309515299036, −2.223261417359666767483609734387, −0.78948036246897909633060233007, 1.425101158665970858539167850356, 1.98051214011085071492843140807, 2.68188034840651906315459658603, 4.28450748679461924404471436722, 5.29451881399247302984269714109, 6.66546821662014930698164973063, 7.26681857386180097739726291581, 8.658494730127003846442163411021, 8.93524872110525244301329632314, 10.22049807823672904041413699578, 11.00671973777927445682623809639, 12.058957261787096272203105680381, 12.42609648522205219126157858171, 13.64628705318484799982565124298, 14.09157391055354425470356614336, 15.05607326705277510895879722145, 16.77018266121403075801808709391, 17.60524496792379373702707537756, 17.80127908493435766096300287467, 18.653360768520680679224664897121, 19.49836903567196697918050083667, 20.24964002595225926265972929421, 21.132452686407615972962297919260, 21.93244949056286249650231829555, 22.49576625164144400115420057925

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