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Properties

Label 1-304-304.163-r1-0-0
Degree $1$
Conductor $304$
Sign $0.904 - 0.427i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.904 - 0.427i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.904 - 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8323302792 - 0.1868442768i\)
\(L(\frac12)\) \(\approx\) \(0.8323302792 - 0.1868442768i\)
\(L(1)\) \(\approx\) \(0.7014663967 + 0.1060590985i\)
\(L(1)\) \(\approx\) \(0.7014663967 + 0.1060590985i\)

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

−24.58196294077175685376198798753, −24.09930291422522952385116680858, −23.80319076824563260026513270984, −22.45922630612905350973156991532, −21.76066882038072618935196578100, −20.68735271899104521119647390343, −19.56612044986492432416148023582, −18.85668550992778014618658770368, −17.87888249593105965571558724610, −16.87026191734696085670292198767, −16.36628759828031060828875813226, −15.142403943556150677685045293039, −14.1247202257352741488382872446, −12.89884968809638302796608831311, −12.10612601068430487404822333197, −11.28325602900172735565426843846, −10.64892250484787138973221544383, −8.91109308123028751436809399079, −8.0084964250062145126845836552, −7.1733326334308737713330847238, −5.8857349919074823085323877116, −4.88259351126387549111947255502, −4.00085712236944450087722672153, −2.08231522302549706449957411961, −0.83428953914328289557100276238, 0.39868222522611643486586789888, 2.18947618130091834929742666426, 3.83123974931393321119633828553, 4.68400218325918409585240604749, 5.58515999367586180595418092297, 7.1597324837713587797359503766, 7.613668332630729657845663391173, 9.21134194213243788966710462657, 10.2490217046562602906261734244, 11.20260812584208675362912432691, 11.794887909676117627754687608945, 12.69806810894007843377153027923, 14.34811756739200691060461080667, 15.13716449801215906875968356590, 15.73600051531521836798683072209, 16.94710675944204427704385208877, 17.81890906049216817863793758596, 18.3754221983096107926406234890, 19.82860888000599973418805575828, 20.50576767460851898745823631401, 21.64308591681003859023197776619, 22.45709668550106123062054055713, 23.141052882393056486650057871198, 23.95298808159222645767326955473, 24.84408551289948906364873480532

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