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Properties

Label 1-304-304.213-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.0917 + 0.995i$
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  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)5-s + (0.5 + 0.866i)7-s + (−0.766 + 0.642i)9-s + (−0.866 − 0.5i)11-s + (−0.342 + 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.642 + 0.766i)21-s + (−0.173 − 0.984i)23-s + (0.939 + 0.342i)25-s + (−0.866 − 0.5i)27-s + (−0.642 − 0.766i)29-s + (−0.5 − 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (0.984 + 0.173i)5-s + (0.5 + 0.866i)7-s + (−0.766 + 0.642i)9-s + (−0.866 − 0.5i)11-s + (−0.342 + 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.642 + 0.766i)21-s + (−0.173 − 0.984i)23-s + (0.939 + 0.342i)25-s + (−0.866 − 0.5i)27-s + (−0.642 − 0.766i)29-s + (−0.5 − 0.866i)31-s + (0.173 − 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.0917 + 0.995i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.0917 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063222320 + 1.165745330i\)
\(L(\frac12)\) \(\approx\) \(1.063222320 + 1.165745330i\)
\(L(1)\) \(\approx\) \(1.172786257 + 0.6120431569i\)
\(L(1)\) \(\approx\) \(1.172786257 + 0.6120431569i\)

Euler product

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

−25.22725272618765368063858827738, −24.195793892814865598905672438731, −23.47353690542457345179625983787, −22.60091783365232717445327017219, −21.2310306596157726871229390097, −20.518230441288955298800033851059, −19.852869800076009878452384492033, −18.569113520516345057079815995819, −17.72583131493617757591253013190, −17.37114146784419148472106831257, −16.044069623414634525770078124266, −14.587832794201902551292476929898, −14.04588513622727647332213741869, −13.0403052609739379695901890112, −12.51395145098093212097970898493, −11.04784670592907021960485378649, −10.08141406111684436308809299927, −9.0683207475836409275213719238, −7.6792173514085533697545772662, −7.33461389847841129237232456181, −5.833384595542688395261135069145, −5.0301042609300713628049558232, −3.267563550158058336353288183018, −2.13463209858264015368557153950, −1.03365252201268507436385453545, 2.0339326745547568485367778070, 2.80458639879322718553645010484, 4.29280853406165230455618425610, 5.4059046247162250414114648978, 6.06211626290554930560267901565, 7.815963543972913371607801378366, 8.80258513688432305524261268998, 9.60639229886964605303641219064, 10.51348726324540269570099028089, 11.41124497007317386961846815180, 12.70654189316545905191445096869, 13.89897709711197139854756807415, 14.56402131395757196361314520248, 15.39518700946583587757633025434, 16.50596993578854551977784953718, 17.208877852539954250667681208790, 18.48112235900607026563892875934, 19.05967189170330055673476285603, 20.62152722251073016023029019101, 21.13248671140056086082655356924, 21.78333341775777010542300838597, 22.50172054484331052130182463660, 23.960056837161906319649117570709, 24.74514894527345583332241913566, 25.84649566381962757272112432213

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