↑

Properties

Label 1-1968-1968.371-r0-0-0
Degree $1$
Conductor $1968$
Sign $0.245 + 0.969i$
Analytic cond. $9.13935$
Root an. cond. $9.13935$
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  + (−0.951 − 0.309i)5-s + (−0.587 + 0.809i)7-s + (−0.309 − 0.951i)11-s + (−0.809 + 0.587i)13-s + (0.951 − 0.309i)17-s + (−0.809 − 0.587i)19-s + (0.809 − 0.587i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.809 − 0.587i)35-s + (−0.951 − 0.309i)37-s + (0.587 + 0.809i)43-s + (−0.587 − 0.809i)47-s + (−0.309 − 0.951i)49-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)5-s + (−0.587 + 0.809i)7-s + (−0.309 − 0.951i)11-s + (−0.809 + 0.587i)13-s + (0.951 − 0.309i)17-s + (−0.809 − 0.587i)19-s + (0.809 − 0.587i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (0.809 − 0.587i)35-s + (−0.951 − 0.309i)37-s + (0.587 + 0.809i)43-s + (−0.587 − 0.809i)47-s + (−0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1968\)    =    \(2^{4} \cdot 3 \cdot 41\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(9.13935\)
Root analytic conductor: \(9.13935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1968} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1968,\ (0:\ ),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5053971071 + 0.3934776179i\)
\(L(\frac12)\) \(\approx\) \(0.5053971071 + 0.3934776179i\)
\(L(1)\) \(\approx\) \(0.7237318067 + 0.02417653611i\)
\(L(1)\) \(\approx\) \(0.7237318067 + 0.02417653611i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 - iT \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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

−19.7147901551656701023999136332, −19.2140081938491878015364048710, −18.60642102982219874448978122081, −17.36561104989712539635250334889, −17.11537614748455187734449130185, −16.09211438040595279409043011743, −15.431280473065975061822167122330, −14.78487919726936267113356991321, −14.1209663849448174748421349759, −12.95383450681828779094029002732, −12.52751887769303306011690767530, −11.81631978179355109800752186370, −10.69973078672873392631485864698, −10.29140709083763063025119462613, −9.5424927772066248122323218493, −8.39847566469930675103475803713, −7.47072261304134538773717861993, −7.30646054250233065842802028104, −6.266547907133764633133778231421, −5.16722442653934780214061888255, −4.34553745931026010327234895531, −3.54769346529726251695472216385, −2.87028811976736308443893067369, −1.62207660676617081243442667201, −0.30178065225560090803585719330, 0.8027084106961864503241402476, 2.27244686614496981844456452798, 3.072092035056514556745874300617, 3.835732793311977269928908809253, 4.91151680865132358705330120930, 5.50278245548361805463192763681, 6.587166094859792831853729800857, 7.278774334373039552378143619260, 8.230858212720569559893914050040, 8.866649156668449227546772636154, 9.500940359901501799037623350789, 10.600652241555725898175654685346, 11.35077944667223432452379589943, 12.0538307518448580275423630457, 12.67688994140332271725681159705, 13.3423140875919944846419793797, 14.53028034574435934722879572999, 14.99027587075915002622530315338, 15.84989663458241160612993322128, 16.52331404702825026161569696713, 16.87938250405450617716459163225, 18.2158332722191154213392978461, 18.870626958839287152803114575091, 19.32001039838298077850464417050, 19.902988924002533023190708879714

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