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Properties

Label 1-643-643.10-r0-0-0
Degree $1$
Conductor $643$
Sign $0.771 + 0.635i$
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  + (−0.358 − 0.933i)2-s + (0.984 + 0.175i)3-s + (−0.742 + 0.669i)4-s + (0.386 + 0.922i)5-s + (−0.189 − 0.981i)6-s + (−0.0733 + 0.997i)7-s + (0.891 + 0.452i)8-s + (0.938 + 0.345i)9-s + (0.722 − 0.691i)10-s + (−0.0146 − 0.999i)11-s + (−0.848 + 0.529i)12-s + (0.331 + 0.943i)13-s + (0.957 − 0.289i)14-s + (0.218 + 0.975i)15-s + (0.102 − 0.994i)16-s + (−0.566 + 0.824i)17-s + ⋯
L(s)  = 1  + (−0.358 − 0.933i)2-s + (0.984 + 0.175i)3-s + (−0.742 + 0.669i)4-s + (0.386 + 0.922i)5-s + (−0.189 − 0.981i)6-s + (−0.0733 + 0.997i)7-s + (0.891 + 0.452i)8-s + (0.938 + 0.345i)9-s + (0.722 − 0.691i)10-s + (−0.0146 − 0.999i)11-s + (−0.848 + 0.529i)12-s + (0.331 + 0.943i)13-s + (0.957 − 0.289i)14-s + (0.218 + 0.975i)15-s + (0.102 − 0.994i)16-s + (−0.566 + 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.771 + 0.635i$
Analytic conductor: \(2.98607\)
Root analytic conductor: \(2.98607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (0:\ ),\ 0.771 + 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.527527177 + 0.5480283624i\)
\(L(\frac12)\) \(\approx\) \(1.527527177 + 0.5480283624i\)
\(L(1)\) \(\approx\) \(1.237084518 + 0.03869560754i\)
\(L(1)\) \(\approx\) \(1.237084518 + 0.03869560754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.358 - 0.933i)T \)
3 \( 1 + (0.984 + 0.175i)T \)
5 \( 1 + (0.386 + 0.922i)T \)
7 \( 1 + (-0.0733 + 0.997i)T \)
11 \( 1 + (-0.0146 - 0.999i)T \)
13 \( 1 + (0.331 + 0.943i)T \)
17 \( 1 + (-0.566 + 0.824i)T \)
19 \( 1 + (0.998 - 0.0586i)T \)
23 \( 1 + (-0.613 + 0.789i)T \)
29 \( 1 + (-0.0146 - 0.999i)T \)
31 \( 1 + (0.218 - 0.975i)T \)
37 \( 1 + (-0.780 - 0.625i)T \)
41 \( 1 + (-0.904 + 0.426i)T \)
43 \( 1 + (-0.989 + 0.146i)T \)
47 \( 1 + (-0.0733 + 0.997i)T \)
53 \( 1 + (0.590 - 0.807i)T \)
59 \( 1 + (0.984 - 0.175i)T \)
61 \( 1 + (0.798 - 0.601i)T \)
67 \( 1 + (0.275 + 0.961i)T \)
71 \( 1 + (-0.465 + 0.884i)T \)
73 \( 1 + (0.798 - 0.601i)T \)
79 \( 1 + (-0.465 - 0.884i)T \)
83 \( 1 + (0.218 + 0.975i)T \)
89 \( 1 + (-0.358 + 0.933i)T \)
97 \( 1 + (0.439 + 0.898i)T \)
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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

−23.04144560691343351859941245493, −22.18346380963541746790076048128, −20.76833462692786972926808564461, −20.01654093664645145491335349294, −19.96206881879960338559726508424, −18.31928873647217288291086814525, −17.93996877692795422120547002665, −16.9821831863896101526222372492, −16.07024417303016582482129296025, −15.4835819767939372360886559362, −14.40901024168725584931088534824, −13.68541662112283322562529781124, −13.17162179414416643713223815590, −12.21753209292051078298488289806, −10.27514237254272825811591904457, −9.961908274470052469185156765, −8.888370851815523486542261720065, −8.27942129762994834009275910936, −7.29895565647731480440620830771, −6.73945674355137125843663504488, −5.230840259361211918032889132577, −4.56476164881246383060494867970, −3.44874825891135351753510118986, −1.81749254930360918933576802317, −0.84230080988568850423708540829, 1.69959701733295591531833271950, 2.37121681916144440421615141913, 3.30268274563770616047861972928, 3.97574835946456741546965401884, 5.48100005864317400272917164450, 6.64482724367764161326123146313, 7.9157481349781710156314406287, 8.62200635423250235135788353824, 9.47831939125386547859645644907, 10.04614525774033510561032692535, 11.23583531066119403081610201934, 11.730914193949582370277736968233, 13.13802987224623834105990408037, 13.68505349923690413922456197574, 14.42674793511885190432321855686, 15.42802271845979091948654867820, 16.29242495056046937973488514139, 17.59852516544979555003362174163, 18.3940960452245128833019631633, 19.09035212613668428197990871037, 19.37830117462261932094401062568, 20.61723019402775169368091623906, 21.417469958309210778713931700153, 21.83098473051926391289743948643, 22.44553993539015738577113516834

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