Properties

Label 1-304-304.293-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: $\chi_{304} (293, \cdot )$
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

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

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