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Properties

Label 1-304-304.61-r0-0-0
Degree $1$
Conductor $304$
Sign $0.769 - 0.639i$
Analytic cond. $1.41177$
Root an. cond. $1.41177$
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  + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)5-s + (0.5 + 0.866i)7-s + (−0.766 + 0.642i)9-s + (0.866 + 0.5i)11-s + (0.342 − 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (0.642 − 0.766i)21-s + (−0.173 − 0.984i)23-s + (0.939 + 0.342i)25-s + (0.866 + 0.5i)27-s + (0.642 + 0.766i)29-s + (−0.5 − 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)5-s + (0.5 + 0.866i)7-s + (−0.766 + 0.642i)9-s + (0.866 + 0.5i)11-s + (0.342 − 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (0.642 − 0.766i)21-s + (−0.173 − 0.984i)23-s + (0.939 + 0.342i)25-s + (0.866 + 0.5i)27-s + (0.642 + 0.766i)29-s + (−0.5 − 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.769 - 0.639i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.769 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9572279609 - 0.3458786970i\)
\(L(\frac12)\) \(\approx\) \(0.9572279609 - 0.3458786970i\)
\(L(1)\) \(\approx\) \(0.8933789477 - 0.2087666513i\)
\(L(1)\) \(\approx\) \(0.8933789477 - 0.2087666513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.342 - 0.939i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.984 - 0.173i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−25.65736739371770365140563350660, −24.27595749836900562888861111861, −23.39569792637634742746721999720, −22.92649887925727540842348725261, −21.77043762668441582026865902426, −21.02235167529192181502670983721, −20.04184504445275351393412502879, −19.329724330521798100423535015922, −18.10413268759653841255068702580, −16.9050506619645457390085982115, −16.41696734972973819758318469690, −15.47950943516747894950516333605, −14.43930418954079198707543748751, −13.81685855802088446476137161647, −11.935847211633805927649293080600, −11.51333470592012920229677844439, −10.64698153290027354707967735131, −9.539034644838956565884665872334, −8.5065633239427297665399312214, −7.38698509364114349596515777550, −6.2967887883304177435630905286, −4.86875871610292375316620328548, −4.0287575359373122104884539524, −3.29371316844469583453775292756, −1.05723827837445215848658732516, 0.9817813875924440765776142837, 2.30916555749355112477508796005, 3.735009783838005307035069012535, 5.09962386586304735022343462956, 6.0754997688063595339287768082, 7.25051824627347008172670971716, 8.146071457259381501121246005056, 8.841722692730227684524403585785, 10.55513448893556678678654869322, 11.55149379537708728134091726486, 12.30168808424074711961688365455, 12.8146822597102834024732874373, 14.37852326067156694688558523950, 15.01787613829430668367944304020, 16.18302705240420659324676595861, 17.16493544011344304299552630205, 18.11850595176805756726685390894, 18.82359910660885216412604042488, 19.73693535600654042695227023710, 20.46884820867954064996743487007, 21.857469173663047084826126054691, 22.77022181707367852526912624471, 23.41585830813599829720160291327, 24.41710868481091417200522853288, 24.953196258982725531644972550005

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