Properties

Label 1-647-647.221-r0-0-0
Degree $1$
Conductor $647$
Sign $0.112 - 0.993i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.932 + 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.850 + 0.526i)5-s + (0.0922 − 0.995i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s + (−0.982 + 0.183i)10-s + (−0.850 − 0.526i)11-s + (0.445 − 0.895i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.739 + 0.673i)15-s + (0.0922 + 0.995i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (−0.273 − 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.850 + 0.526i)5-s + (0.0922 − 0.995i)6-s + (0.445 − 0.895i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s + (−0.982 + 0.183i)10-s + (−0.850 − 0.526i)11-s + (0.445 − 0.895i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.739 + 0.673i)15-s + (0.0922 + 0.995i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ 0.112 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217047995 - 1.087268614i\)
\(L(\frac12)\) \(\approx\) \(1.217047995 - 1.087268614i\)
\(L(1)\) \(\approx\) \(1.332265167 - 0.3114676968i\)
\(L(1)\) \(\approx\) \(1.332265167 - 0.3114676968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (0.932 + 0.361i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
7 \( 1 + (0.445 - 0.895i)T \)
11 \( 1 + (-0.850 - 0.526i)T \)
13 \( 1 + (0.445 - 0.895i)T \)
17 \( 1 + (-0.273 - 0.961i)T \)
19 \( 1 + (0.0922 + 0.995i)T \)
23 \( 1 + (0.0922 - 0.995i)T \)
29 \( 1 + (0.0922 - 0.995i)T \)
31 \( 1 + (-0.850 - 0.526i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (0.445 + 0.895i)T \)
43 \( 1 + (0.932 - 0.361i)T \)
47 \( 1 + (-0.273 - 0.961i)T \)
53 \( 1 + (0.0922 - 0.995i)T \)
59 \( 1 + (-0.982 + 0.183i)T \)
61 \( 1 + (-0.273 + 0.961i)T \)
67 \( 1 + (-0.273 + 0.961i)T \)
71 \( 1 + (0.0922 - 0.995i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (0.932 - 0.361i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (-0.982 + 0.183i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.08125203384461684535571033056, −22.02340430006625036735526843871, −21.497873676602443191767207307, −20.844060295099003587049482329839, −20.05232199568323277616949869406, −19.292721160317595527163483171743, −18.20761118260210506328973013376, −17.053366807786124749727183426724, −15.94144020223002881748415106995, −15.571916626332245774174008817218, −14.9833328630288896041997958471, −13.98873527629207453168098355541, −12.73857620960101495382905165737, −12.18414123803550192761411139475, −11.12844990398435399693846375698, −10.927379462180120126854630094, −9.466393920516409464058681681833, −8.772281649993840578405838296926, −7.541338399481966726793835269336, −6.2591211487194899183713620607, −5.210308933978785452697672104530, −4.73289262573034577407857703360, −3.84639467928728183239921657186, −2.859888809922093546017303429888, −1.56627622989316250900883328892, 0.60351449111619664291522361927, 2.29325469112349625569193262172, 3.211183748733638770122635131756, 4.21769611523727213342433120720, 5.341524312657144555226803127473, 6.20745370150875122668750878482, 7.220333964225160560746687028170, 7.78362657696462690926142651892, 8.320138711453534878954957456275, 10.55582184320545494879163173851, 11.0465336857326497668112624722, 11.842236493864695019273716690066, 12.77337942656179582828328879819, 13.49018867134532210463999364237, 14.22734389467200872708576713127, 15.014856024337297701862246512116, 16.129798259321762874558260874009, 16.627662212263023786876812747746, 17.86712206506834270904558807857, 18.40986016368668315039833737899, 19.5138398382754842890362748804, 20.35578559191349053328338361461, 20.93380388231210481341904670036, 22.34379643950372090354029959872, 23.05914988923553683882699001774

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