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Properties

Label 1-304-304.275-r1-0-0
Degree $1$
Conductor $304$
Sign $-0.231 + 0.972i$
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  + (−0.984 − 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

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

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.231 + 0.972i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ -0.231 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03463805970 + 0.04386379431i\)
\(L(\frac12)\) \(\approx\) \(0.03463805970 + 0.04386379431i\)
\(L(1)\) \(\approx\) \(0.6104261867 - 0.1998124632i\)
\(L(1)\) \(\approx\) \(0.6104261867 - 0.1998124632i\)

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

−24.885367227471435174911690490122, −23.84247754382941948925573262011, −22.81323134803260227796647084087, −22.13722493803839031128725354621, −21.65303733702014077300230469915, −20.541920991121158008732201544864, −19.10935238184441007743583871954, −18.36097542947498459636716158291, −17.64805753677524783967992498702, −16.82385284060498504827441569741, −15.40098890258181784785165225170, −15.25363665717557658851468113673, −13.6275804491827702982083699991, −12.71032010303242631013811840380, −11.83201139450374055719554825614, −10.748209406829859191151077867898, −9.98477821979307346093706048831, −9.14665802689161629176992763885, −7.37357715628994683917889937521, −6.59416085007671238012684952897, −5.55557950099696569752653871474, −4.81272668575440035077146731431, −3.04516890299077602305093623834, −2.03296356433190314620781407568, −0.021587120726931662639335034507, 1.00939271155775938162670315672, 2.49292814884456510747926938963, 4.33426192381192671353636063349, 5.08843911728420213418090899324, 6.19982805457172986465781242126, 7.06132490877067773638820258062, 8.311146602658896341506902501463, 9.68629336277900745658431233885, 10.38525305991093410461805176468, 11.35482855839850847731564188524, 12.69908198359446027487615989790, 13.052525859907306269955239712696, 14.08467894289448420304212551110, 15.66973042754606649379383279487, 16.45094194397602709354884784053, 17.18262278326314846536041427154, 17.80230423265579103117472384198, 19.021381493459667641962846405676, 19.93325737053236412037742910387, 21.07591897181995868713757740695, 21.75315094024339820395993831893, 22.75921564493721650299988299293, 23.60626973550286085907174940866, 24.35067844542115718137007883526, 25.08703925419285229080770692766

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