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Properties

Label 1-304-304.221-r1-0-0
Degree $1$
Conductor $304$
Sign $-0.730 + 0.683i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
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  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s − 7-s + (0.5 + 0.866i)9-s − i·11-s + (−0.866 + 0.5i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)21-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − i·27-s + (−0.866 + 0.5i)29-s − 31-s + (0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s − 7-s + (0.5 + 0.866i)9-s − i·11-s + (−0.866 + 0.5i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)21-s + (0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − i·27-s + (−0.866 + 0.5i)29-s − 31-s + (0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.730 + 0.683i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ -0.730 + 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7061596716 + 1.787710552i\)
\(L(\frac12)\) \(\approx\) \(0.7061596716 + 1.787710552i\)
\(L(1)\) \(\approx\) \(1.188330689 + 0.5491493068i\)
\(L(1)\) \(\approx\) \(1.188330689 + 0.5491493068i\)

Euler product

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

−24.88187358114918004159999794827, −24.220357231398535234828597365687, −22.85570194245001172220234481564, −22.1594262611669641498550047423, −20.85909142961018540669390960754, −20.268928273476979309736385254961, −19.50438603302987213705058523214, −18.4216286308354594906654491411, −17.62441474031421603394237092707, −16.62747538122260418317578470601, −15.475756660599975756340018709055, −14.57031039783126485161475093884, −13.57319298575228685438555492562, −12.77419423074482015479597066141, −12.29560612489209454013820871290, −10.39882215282225911730669696034, −9.44015711768853093684415787238, −9.035907349206777193523336260474, −7.50352032365875657677262695417, −6.79874403017899606496717483183, −5.54606844280068978081201981830, −4.24952374453328368590324856708, −2.77725862262860058988103818946, −2.04408929246005791845657781746, −0.459984253521653519338379643797, 1.79806275160313932640148096596, 2.92699727089971148544965009746, 3.70560448747802418550735451727, 5.22714785524714124986791912335, 6.35779898181080468835961584047, 7.360770029673710125049223120, 8.76716131632485617380189045024, 9.486100883791160886290616055798, 10.26136011698323517050583023575, 11.24875073123144259264758433601, 12.950074326819739943018506235643, 13.47064906724924975526638309277, 14.478385392657118234659151434641, 15.20424699303922668775264581305, 16.38091543500041584489748501824, 17.04162648828635637089657649615, 18.46658845750231355531491180143, 19.24310213872861863386296776899, 19.89599311961116655974400302430, 21.1564898597710837432042970736, 21.89278916295860926793940068539, 22.26723835160755190865806118837, 23.79802671145311677148913682373, 24.7859011381972363737037068029, 25.605091852467401671680417153301

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