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Properties

Label 1-304-304.125-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.427 - 0.904i$
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  + (0.866 − 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

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

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.427 - 0.904i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.427 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4930851711 - 0.7785449391i\)
\(L(\frac12)\) \(\approx\) \(0.4930851711 - 0.7785449391i\)
\(L(1)\) \(\approx\) \(0.8961081563 - 0.3364838958i\)
\(L(1)\) \(\approx\) \(0.8961081563 - 0.3364838958i\)

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

−25.77648550833304275638285927621, −24.794951264393625671520170885403, −23.90449989256486658823308289673, −22.84806117222702048130358078157, −22.02289193205048619985719717489, −20.999613018003556241722411034927, −19.95288147336182659089328491447, −19.609140265779837536084297705870, −18.78224475239284229126212171245, −17.16008354984966117879439988982, −16.40411488874761816833044096705, −15.27515881456362315959880637922, −15.070938558559051509319011801168, −13.56954623902956788648115710427, −12.75860043560081654755821772122, −11.86242344001868595969579251222, −10.44478691014852872008345981641, −9.54265341907169216466330407903, −8.804768383114283271645168994950, −7.64989603020844815902564444819, −6.839530041794076006569718766485, −5.04720941355947441664195706125, −4.14312728804428679361603208794, −3.24180327267290910140910541670, −1.902335051113215406212142852617, 0.521523323773274458620682481193, 2.674404232240371205897591540993, 3.14597852288007307685661632341, 4.381548585505095588668920566177, 6.16160067229966466645308597162, 7.07154079535274245421411720392, 7.89439389164788527680944767398, 8.926578089459351932782310643287, 9.9035881541273189151780509660, 11.13178541644805300816880225056, 12.21856214428218372144347775240, 13.06410064493141559001886153996, 13.99976893498180654993938578822, 14.96783693075525300886667733269, 15.73501086789353858728048979853, 16.684533578406782872919089372450, 18.16340622452977116317450096268, 18.91193127511376108804348189335, 19.57151095246113234406119886831, 20.18444774144516924884183495666, 21.43254292121395580215331819748, 22.562360034150116164372794337791, 23.13695979614415508728237190597, 24.49098369075816499192025506363, 24.77258322763319056149898075651

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