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Properties

Label 1-675-675.556-r0-0-0
Degree $1$
Conductor $675$
Sign $0.280 - 0.959i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.939 − 0.342i)7-s + (0.669 − 0.743i)8-s + (−0.241 + 0.970i)11-s + (−0.719 + 0.694i)13-s + (0.438 + 0.898i)14-s + (−0.997 + 0.0697i)16-s + (−0.978 − 0.207i)17-s + (0.669 − 0.743i)19-s + (0.848 − 0.529i)22-s + (0.559 − 0.829i)23-s + 26-s + (0.309 − 0.951i)28-s + (0.990 + 0.139i)29-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.939 − 0.342i)7-s + (0.669 − 0.743i)8-s + (−0.241 + 0.970i)11-s + (−0.719 + 0.694i)13-s + (0.438 + 0.898i)14-s + (−0.997 + 0.0697i)16-s + (−0.978 − 0.207i)17-s + (0.669 − 0.743i)19-s + (0.848 − 0.529i)22-s + (0.559 − 0.829i)23-s + 26-s + (0.309 − 0.951i)28-s + (0.990 + 0.139i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.280 - 0.959i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (556, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5373497661 - 0.4029649800i\)
\(L(\frac12)\) \(\approx\) \(0.5373497661 - 0.4029649800i\)
\(L(1)\) \(\approx\) \(0.6102332567 - 0.1938204135i\)
\(L(1)\) \(\approx\) \(0.6102332567 - 0.1938204135i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (-0.719 + 0.694i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.559 - 0.829i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
31 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.615 + 0.788i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (0.990 - 0.139i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (0.990 + 0.139i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−23.01098864839170404003624789128, −22.254215912226930979019839950397, −21.39295383319177271276545130529, −20.01241553232511049453725806363, −19.60389999287398996882998569995, −18.733475079519882938353852379852, −18.0117539697136503330646258589, −17.120576088525887416125744685878, −16.284211481874055659256561634226, −15.67910636379808362548134446331, −14.9273099874873468667106308911, −13.85330673064469916737050594772, −13.10932075354366689795271171049, −11.98021554885621244943658723108, −10.88462484274153272101762321675, −10.0962249442964716105284410725, −9.27624093105025149902906817865, −8.478388902795746620659193070907, −7.56193180296794797484001077183, −6.61555407834799876945916932153, −5.79876882231881015443342169796, −5.0212642894993215872033326764, −3.457901856909582217595915412953, −2.413004685916541376531877608301, −0.87786039157637163373634709014, 0.55905782147979977704076846588, 2.12727311146668708484002831133, 2.827218015260251346104810424518, 4.074429870062607607706777526384, 4.876423744326511563284681171, 6.7364908752606271387340314442, 7.040327658158109511486318141366, 8.2187219979498165002053316781, 9.47749524203768054911771627132, 9.613507623155813427403948293419, 10.75923135315751466977873416857, 11.560137101959287301964996754262, 12.597781712500265306006701285164, 13.05649376447284880971678000698, 14.125578217467135469097915548431, 15.392248878233089863953237800073, 16.183952675105031186485263451029, 16.98130893724448627396181685940, 17.7511005075399193725414120431, 18.54671723829595839525647248489, 19.46656793882308744988055193087, 20.019721585886852694855741691207, 20.67510765339233765655938696835, 21.831500965471849967053201798903, 22.33534158489871598634648600532

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