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Properties

Label 1-304-304.219-r0-0-0
Degree $1$
Conductor $304$
Sign $0.0204 - 0.999i$
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  + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)5-s + (−0.5 − 0.866i)7-s + (0.939 + 0.342i)9-s + (0.866 + 0.5i)11-s + (−0.984 + 0.173i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.342 + 0.939i)21-s + (0.766 − 0.642i)23-s + (−0.173 − 0.984i)25-s + (−0.866 − 0.5i)27-s + (0.342 − 0.939i)29-s + (−0.5 − 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)5-s + (−0.5 − 0.866i)7-s + (0.939 + 0.342i)9-s + (0.866 + 0.5i)11-s + (−0.984 + 0.173i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (0.342 + 0.939i)21-s + (0.766 − 0.642i)23-s + (−0.173 − 0.984i)25-s + (−0.866 − 0.5i)27-s + (0.342 − 0.939i)29-s + (−0.5 − 0.866i)31-s + (−0.766 − 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0204 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0204 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.0204 - 0.999i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.0204 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3336283871 - 0.3405272100i\)
\(L(\frac12)\) \(\approx\) \(0.3336283871 - 0.3405272100i\)
\(L(1)\) \(\approx\) \(0.5992784500 - 0.08180036177i\)
\(L(1)\) \(\approx\) \(0.5992784500 - 0.08180036177i\)

Euler product

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

−25.24120782348702535089952795118, −24.54634741444715799648849090414, −23.80480344648474764115670743082, −22.76464684690757882603186312129, −22.03797116551967569742058599542, −21.36859222559679282148077039913, −19.971143401734895836944483969855, −19.33102033891214805181471613819, −18.23632160383560425969006767091, −17.20794888235542712235504513738, −16.464845241959228191847492271675, −15.69649264429704115558685006823, −14.878809048522478052266101928342, −13.26331225417945344222192202658, −12.33030706389450041975700962284, −11.815916095787517800711852586079, −10.85760694363382371024254252394, −9.438996164029246526149577783104, −8.864050669659372065188987493, −7.356944985707755279074608728371, −6.34615222022581882983078223211, −5.25014377592761026001596798334, −4.47065344487016136604988648601, −3.1124775102202068168722132197, −1.250891067390067666755899135654, 0.392419860102968147245078810508, 2.21659522210304486889750817659, 3.889722717488447594637630967601, 4.56410086129647127298808250040, 6.19262954145505620854820752840, 6.97026141590120349969558895104, 7.55245329805351243229072605395, 9.33408916968187111856912557321, 10.39588881100348022310270452292, 11.08742917203114336395113978720, 12.053233166325926961475889845302, 12.870258372651153806814553530915, 14.111093433264776373379580136854, 15.115354631813933760660457869284, 16.0607613934395960298599436317, 17.13187261797411587872730934136, 17.55622359461849232588411199548, 18.94512100173081398360761159631, 19.4336581284736372311381512554, 20.48806851634078997586327183154, 22.04061460250561854432362396873, 22.426321873322174386539750907797, 23.17987309715703969235290229244, 24.008940632416102121002337925915, 24.91299128738525894998635564736

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