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Properties

Label 1-304-304.269-r1-0-0
Degree $1$
Conductor $304$
Sign $-0.843 - 0.537i$
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  + (−0.642 − 0.766i)3-s + (−0.342 + 0.939i)5-s + (0.5 − 0.866i)7-s + (−0.173 + 0.984i)9-s + (−0.866 + 0.5i)11-s + (0.642 − 0.766i)13-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)17-s + (−0.984 + 0.173i)21-s + (0.939 − 0.342i)23-s + (−0.766 − 0.642i)25-s + (0.866 − 0.5i)27-s + (−0.984 − 0.173i)29-s + (0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (−0.342 + 0.939i)5-s + (0.5 − 0.866i)7-s + (−0.173 + 0.984i)9-s + (−0.866 + 0.5i)11-s + (0.642 − 0.766i)13-s + (0.939 − 0.342i)15-s + (0.173 + 0.984i)17-s + (−0.984 + 0.173i)21-s + (0.939 − 0.342i)23-s + (−0.766 − 0.642i)25-s + (0.866 − 0.5i)27-s + (−0.984 − 0.173i)29-s + (0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.843 - 0.537i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ -0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1694708264 - 0.5807225454i\)
\(L(\frac12)\) \(\approx\) \(0.1694708264 - 0.5807225454i\)
\(L(1)\) \(\approx\) \(0.7223372891 - 0.1692705277i\)
\(L(1)\) \(\approx\) \(0.7223372891 - 0.1692705277i\)

Euler product

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

−25.40686465980561713121308678955, −24.4229438580607635706860595635, −23.622420962262532928117323160693, −22.89816702895720418729963211053, −21.60436539599363559961394942573, −21.1049238784353350840667552050, −20.454889084551415761376572972077, −19.01729421470305576330160029956, −18.17546279136768142032889372387, −17.15999508823276097864177573617, −16.08813634796771825839974040378, −15.83397960098662902104626398700, −14.68444450323940136289071128879, −13.39661424344659692941569936534, −12.27551016217024049321921654269, −11.518504193774607427652843666821, −10.76133364183978010115337464234, −9.23225297401271297032198644465, −8.85846842422661795172599284810, −7.53578240996795306394759284355, −5.92021232002940420393288280412, −5.1943807226109539398021203708, −4.37839143855609051827787093334, −3.01212150621158539515511856640, −1.23345224231288149358403025596, 0.21523677454984465161297743078, 1.61577835508983750608772698036, 2.9748653909146725636610793623, 4.34610121078386388291631825871, 5.61912231626811445684652366634, 6.65766835040550244132529677873, 7.60282880309394423803159657302, 8.145555271048856460889382356317, 10.21320674192727814917429964333, 10.78339650690368168930290555552, 11.53182594829788288011511231065, 12.842034988781371023669492843480, 13.45774619864771525095053612250, 14.662905266444731126746257371305, 15.491404265667929273864191000, 16.79851944488392314691191815769, 17.58608792889282604838445261537, 18.36520450635686197960152538766, 19.08047697876418973266230840934, 20.16054440480897290906471086058, 21.12762250038301722584269069249, 22.41610258356901047391099318921, 23.08609232048524467940188606279, 23.60646331287774663065574709194, 24.53859467333824411838457055826

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