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Properties

Label 1-304-304.237-r0-0-0
Degree $1$
Conductor $304$
Sign $0.972 + 0.231i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.972 + 0.231i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.972 + 0.231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.002236422 + 0.2353078418i\)
\(L(\frac12)\) \(\approx\) \(2.002236422 + 0.2353078418i\)
\(L(1)\) \(\approx\) \(1.613989329 + 0.1015644536i\)
\(L(1)\) \(\approx\) \(1.613989329 + 0.1015644536i\)

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

−25.24896131885218411094678184990, −24.54261301725962667843020620134, −23.75275997481746999215873077108, −22.307765221888415147911959568977, −21.80784717654456828782375442224, −20.63378466695013931361959378136, −19.91364512599160682525099766515, −19.112943352030215023009945020655, −18.01414023253354727146339550589, −17.35701203494621305896175428115, −16.12464350884212480546935018701, −14.79214182114831295909043730131, −14.26096806863964469364755927807, −13.67733518185630353378022252133, −12.521389836865434518004267018621, −11.1643674936033230621045502818, −10.23014999618342163119084448003, −9.34544966681603189719221130213, −8.270392492871639244415740797279, −7.140495997874810961147314675504, −6.535681154771819113276271965874, −4.817304679767231657609578201957, −3.658126589837528278547794358564, −2.57850429775204221431844316671, −1.46976944706219825581550934304, 1.78351923760303978753739550387, 2.33256815197859557269887051371, 4.06467783805031469059946567817, 4.898944190910744111030160079412, 6.15543518001286975947369675013, 7.53634305673234688455553456846, 8.582915252406065637453506481799, 9.31281578227166017213954045746, 9.965511572578000957874385048721, 11.607866692613511185971977997866, 12.526471624231699516132703472, 13.44241308008846474498815399483, 14.45457478900011470746351397054, 15.09873541996689572290366451480, 16.13268730440212140996791689062, 17.31073032753019569564642656034, 18.01081508239026567204877320426, 19.33796314569477473619735437546, 19.94365480610580405459001095994, 20.881519174927052300827639212580, 21.68210838544061959298181836105, 22.26621008660710070053013716809, 24.07044261295297038545501818918, 24.60333683179691749469808377260, 25.214490914575637160119866189501

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