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Properties

Label 1-304-304.155-r0-0-0
Degree $1$
Conductor $304$
Sign $0.215 + 0.976i$
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  + (−0.642 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.215 + 0.976i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4041592548 + 0.3245890116i\)
\(L(\frac12)\) \(\approx\) \(0.4041592548 + 0.3245890116i\)
\(L(1)\) \(\approx\) \(0.6842360550 + 0.02333321266i\)
\(L(1)\) \(\approx\) \(0.6842360550 + 0.02333321266i\)

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

−25.3644043118604300811773713924, −23.98657263246723491655997963793, −23.07949823601133316286964450930, −22.47677234508725507937531433354, −21.74030331256143455819245398463, −20.73633203454399260539366073146, −19.88311543405781264208337525819, −18.57406650565926748434159218298, −17.85625134956261672613377367550, −16.895393890055819055328541575080, −16.04689776832607981352333751483, −15.19148910976316703279458604011, −14.14825285344104990613558655764, −13.25317177068692493464048153860, −11.95144421580907732096617844858, −10.89629661116328368301914214831, −10.19594289417313530379034393704, −9.6657949922497979596234835978, −7.94359364614569769340642978538, −6.863951949048274134003224136493, −5.913136012755352935084529594061, −4.8849807518476842202297568286, −3.566852482077630003559092414856, −2.68743041741757264136598738848, −0.353781074422993059230314526914, 1.57578382860150414355312819461, 2.50643233648990383649402278906, 4.481415685953920766484920632778, 5.482927180768760426544374150931, 6.205107453866261012265585014, 7.47186553160965094750543411601, 8.4893759545224073428808293964, 9.54726189809818474539590540736, 10.60694123428999067242218260377, 12.08386546188761867618496880678, 12.42190457551860747549596501824, 13.23235041423768637603115600576, 14.36331640408108105994226446785, 15.80585198889832568148263298754, 16.41763757220873606021167253291, 17.50461345791367083253430453901, 18.107177094349404909541653375191, 19.23489313441821739797065076698, 19.86296255463843292625148575798, 21.34041676087218893436681065656, 21.783713484326970814685916127242, 23.02124636234643659576150402359, 23.8332143060605142903203281078, 24.5043060339051902045393214816, 25.37178942186033757296151633269

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