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Properties

Label 1-1968-1968.1397-r0-0-0
Degree $1$
Conductor $1968$
Sign $0.949 + 0.312i$
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  + 5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + (0.707 − 0.707i)17-s + (0.707 + 0.707i)19-s − 23-s + 25-s + (0.707 − 0.707i)29-s − 31-s + (0.707 + 0.707i)35-s + i·37-s + 43-s + (0.707 − 0.707i)47-s + i·49-s + ⋯
L(s)  = 1  + 5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s + (0.707 − 0.707i)17-s + (0.707 + 0.707i)19-s − 23-s + 25-s + (0.707 − 0.707i)29-s − 31-s + (0.707 + 0.707i)35-s + i·37-s + 43-s + (0.707 − 0.707i)47-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1968\)    =    \(2^{4} \cdot 3 \cdot 41\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(9.13935\)
Root analytic conductor: \(9.13935\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1968} (1397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1968,\ (0:\ ),\ 0.949 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.340049939 + 0.3747267969i\)
\(L(\frac12)\) \(\approx\) \(2.340049939 + 0.3747267969i\)
\(L(1)\) \(\approx\) \(1.455690798 + 0.1091065353i\)
\(L(1)\) \(\approx\) \(1.455690798 + 0.1091065353i\)

Euler product

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

−20.059082823050839686861466979812, −19.42765005184958924665931711587, −18.04201602177928848079089731351, −17.876180210892107790679134604842, −17.129173692896627083246341595452, −16.566898558444147755165712600, −15.48232732764255237894613302226, −14.40982131656534766605892133281, −14.38323450128387286267355935063, −13.407956147120155696438120289488, −12.532664542021865276379199153093, −11.979422929547456964278022754130, −10.75235981547940753671773477835, −10.3711562766468472513349276728, −9.52064017323032146264169703780, −8.8649695955750668387261173684, −7.644021357572534132705425352508, −7.28174345161033634141614520894, −6.18180854046232046114350450524, −5.423672231102917318020191163265, −4.658991407070837627808499534261, −3.78243269970224611151792902161, −2.65412498255121536242854662586, −1.75088220673876196317340334031, −0.98383682224879215517465526013, 1.09090007091351023321022197529, 1.94793864524399931058525582872, 2.6918088590409991053131340924, 3.785395319959809748312983814001, 4.85976874964329643704610818882, 5.57817821396804639459827360182, 6.160810539463473277617102071279, 7.13772044015250081901887165486, 8.067170737655332758548735939765, 8.88122058149511746010011678887, 9.584393650051535178547120577617, 10.14336614807159641379127710894, 11.298930429355229629416256120855, 11.889047825875301420955081625022, 12.47429415289652069393637744375, 13.73340340411006013860538749466, 14.15302964617896664174202536372, 14.600342385095841774976445373123, 15.6940776214801895303104757321, 16.57167194522582181562719641183, 17.03821175399214353888466121583, 17.953574828084551420195160462639, 18.48004472441389792925782573514, 19.1308033407293370222885033337, 20.13762263409405681084380049563

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