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Properties

Label 1-1968-1968.1493-r0-0-0
Degree $1$
Conductor $1968$
Sign $0.585 - 0.810i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)5-s + (−0.453 − 0.891i)7-s + (0.156 + 0.987i)11-s + (−0.453 + 0.891i)13-s + (−0.156 − 0.987i)17-s + (0.453 + 0.891i)19-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s + (0.987 + 0.156i)29-s + (0.809 + 0.587i)31-s + (−0.891 − 0.453i)35-s + (−0.587 − 0.809i)37-s + (−0.309 − 0.951i)43-s + (−0.453 + 0.891i)47-s + (−0.587 + 0.809i)49-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)5-s + (−0.453 − 0.891i)7-s + (0.156 + 0.987i)11-s + (−0.453 + 0.891i)13-s + (−0.156 − 0.987i)17-s + (0.453 + 0.891i)19-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s + (0.987 + 0.156i)29-s + (0.809 + 0.587i)31-s + (−0.891 − 0.453i)35-s + (−0.587 − 0.809i)37-s + (−0.309 − 0.951i)43-s + (−0.453 + 0.891i)47-s + (−0.587 + 0.809i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1968\)    =    \(2^{4} \cdot 3 \cdot 41\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(9.13935\)
Root analytic conductor: \(9.13935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1968} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1968,\ (0:\ ),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546472156 - 0.7911972090i\)
\(L(\frac12)\) \(\approx\) \(1.546472156 - 0.7911972090i\)
\(L(1)\) \(\approx\) \(1.162525549 - 0.2236356385i\)
\(L(1)\) \(\approx\) \(1.162525549 - 0.2236356385i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (-0.453 - 0.891i)T \)
11 \( 1 + (0.156 + 0.987i)T \)
13 \( 1 + (-0.453 + 0.891i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.156 - 0.987i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.453 + 0.891i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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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

−19.83311647576128566473595834356, −19.394641017801177666970215471807, −18.64039072149968189251747579472, −17.83710132396039543878996219187, −17.39958384020295300335501468832, −16.47738468153330975193848575654, −15.49685658132375287379679770880, −15.13941572801411086471857986331, −14.19535028447032819989005628153, −13.38704554526664546572641327677, −12.961720852121733163400076370686, −11.83106813277535869240308628892, −11.29888853398489301200753645934, −10.15220115057701974396559041864, −9.89270642436391071279317992700, −8.79541118116240528706710092312, −8.25690204799725921892177529808, −7.090035760293900158066305330745, −6.27800212979717074787545373556, −5.73148923033980906131267390096, −5.02341517910242854499053902862, −3.60170939775666851118774540595, −2.89758736573846787324949813630, −2.251920498577885805247454152831, −1.00724325937769681970581719111, 0.69564665525932706420102502478, 1.744437940637801417204122628433, 2.524975930567620167332898094479, 3.74414534541241695964140990617, 4.62665769470308935942476850390, 5.128476673607207442991850013341, 6.42645966952111087132370532467, 6.83024887875582168843514717325, 7.74610695544122960593812071526, 8.76995347917308620554832132958, 9.60619695717410433608407937380, 9.98731205296871356125184207736, 10.765765422171127985518157687480, 12.15821796304985539655209119008, 12.27412402884747141225979608246, 13.37011924672030837916379503273, 14.07519926942332311155178022601, 14.38476920834989434499179013317, 15.719602341906596858514071409954, 16.372052232543574372896115170090, 16.90657142999230161990836261240, 17.6662497772791504590485628411, 18.24885641397740077359624528278, 19.28947511538101645843578356708, 19.97759983211449311226201363680

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