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Properties

Label 1-320-320.229-r0-0-0
Degree $1$
Conductor $320$
Sign $0.290 + 0.956i$
Analytic cond. $1.48607$
Root an. cond. $1.48607$
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  + (0.382 + 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s − i·17-s + (0.923 − 0.382i)19-s + (−0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (−0.923 − 0.382i)27-s + (0.382 + 0.923i)29-s − 31-s + 33-s + (−0.923 − 0.382i)37-s + (0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (0.923 − 0.382i)13-s − i·17-s + (0.923 − 0.382i)19-s + (−0.382 + 0.923i)21-s + (−0.707 + 0.707i)23-s + (−0.923 − 0.382i)27-s + (0.382 + 0.923i)29-s − 31-s + 33-s + (−0.923 − 0.382i)37-s + (0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.258576971 + 0.9334242984i\)
\(L(\frac12)\) \(\approx\) \(1.258576971 + 0.9334242984i\)
\(L(1)\) \(\approx\) \(1.204773562 + 0.4831991763i\)
\(L(1)\) \(\approx\) \(1.204773562 + 0.4831991763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.923 - 0.382i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + T \)
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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

−24.80766748624087695739646376752, −24.229382651568681480837986079977, −23.16756919386921682849661915351, −22.69065719351884117580638479442, −21.05774812879664538374607593433, −20.38417102251666643782793242480, −19.7606489238732393264851122847, −18.41603500062774101559187778724, −18.03132911950940580786537694400, −17.01747929705543170323014838894, −15.923840944504976664940468865137, −14.58131721998169424761940159394, −14.03389370732641315545811095145, −13.18863961522404359322966940938, −12.01535929829273081283281247502, −11.3717096037652634219194637467, −10.00816751679539490052930174197, −8.92312440746246512179325745431, −7.84798806131000492682422631871, −7.15962796573565354903223587166, −6.137813742500233746988963164926, −4.69776675868976109399978897288, −3.551232253883629661676045226785, −2.10081255172815382002579623338, −1.11321296414924467187572665716, 1.635535112016946328143836176792, 3.11269309042568159837147252365, 3.94348000477298877601521570638, 5.30455921524482955878315935212, 5.95903403378706406057529240075, 7.74983856454591667707887915750, 8.65825179331728013885504231461, 9.25287908683358416029396968675, 10.687874037407470749381264978724, 11.18215198434364774150379697338, 12.37386736019294566273323742734, 13.795044348902768133565863875155, 14.36024158263312944083060557847, 15.52088305191279726031957503224, 15.97039596435035834263844082964, 17.16082348254306654651954986149, 18.104748439695598423948519192924, 19.16345370867101690616538974832, 20.09717345199371417406461920930, 20.96827781189259252246036647920, 21.776475139905256282511603657732, 22.26304686283595691646406247703, 23.648373791512628284478360663341, 24.47883092633107001236224307150, 25.48061758198215762811615558223

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