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Properties

Label 1-299-299.187-r1-0-0
Degree $1$
Conductor $299$
Sign $-0.254 + 0.967i$
Analytic cond. $32.1320$
Root an. cond. $32.1320$
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  + (0.281 + 0.959i)2-s + (−0.654 − 0.755i)3-s + (−0.841 + 0.540i)4-s + (−0.989 + 0.142i)5-s + (0.540 − 0.841i)6-s + (0.909 + 0.415i)7-s + (−0.755 − 0.654i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (0.281 − 0.959i)11-s + (0.959 + 0.281i)12-s + (−0.142 + 0.989i)14-s + (0.755 + 0.654i)15-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (−0.989 + 0.142i)18-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.654 − 0.755i)3-s + (−0.841 + 0.540i)4-s + (−0.989 + 0.142i)5-s + (0.540 − 0.841i)6-s + (0.909 + 0.415i)7-s + (−0.755 − 0.654i)8-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)10-s + (0.281 − 0.959i)11-s + (0.959 + 0.281i)12-s + (−0.142 + 0.989i)14-s + (0.755 + 0.654i)15-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (−0.989 + 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $-0.254 + 0.967i$
Analytic conductor: \(32.1320\)
Root analytic conductor: \(32.1320\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{299} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 299,\ (1:\ ),\ -0.254 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6011452393 + 0.7797285718i\)
\(L(\frac12)\) \(\approx\) \(0.6011452393 + 0.7797285718i\)
\(L(1)\) \(\approx\) \(0.7336894504 + 0.2891319416i\)
\(L(1)\) \(\approx\) \(0.7336894504 + 0.2891319416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.281 + 0.959i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
11 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (0.755 + 0.654i)T \)
37 \( 1 + (-0.989 - 0.142i)T \)
41 \( 1 + (-0.989 + 0.142i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (-0.909 + 0.415i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (0.281 + 0.959i)T \)
71 \( 1 + (0.281 + 0.959i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.989 + 0.142i)T \)
89 \( 1 + (0.755 - 0.654i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−24.50459082754994899696235123221, −23.53733771258980995295137392060, −23.05025358447433741622501502983, −22.2243870376775977226650576073, −21.17426553291586044817204919014, −20.49584014431261295899247989803, −19.81393395795461177629784866217, −18.65350239517591849257411468078, −17.55588962797527389915655342564, −16.94935944815942429097139955114, −15.35785659336229155202023594163, −15.00775901452160705635238110667, −13.78403507007703734884516892307, −12.2729046499155112659951343039, −11.93952338494082230884612784557, −10.844601556874863087249602222112, −10.32486895752109673062442223779, −9.041225518903145408767825502010, −8.02868848745521689778203387636, −6.472059933867064995061244769908, −4.97098251521302497239419900235, −4.369197837330204856213974637784, −3.64384495865617352933486462590, −1.84912228764045014912532029702, −0.42001059389789550964058332222, 0.86059176813328748604772892650, 2.83906686448885606831428542172, 4.42172372307351646560824084589, 5.18037396343202004367895026757, 6.42528990312873536808566199114, 7.147122377651174149638384317466, 8.25355765298599572820299026737, 8.74394531930557416271953589494, 10.85444003272071419879991490949, 11.63162495088356653330801959641, 12.42058881665916772045536488485, 13.57088661463047466871262232358, 14.37589707867258928559193847505, 15.51507051360854273716629896926, 16.174004284409805404995379860709, 17.28135603762764685219537948472, 17.97948053686233391535358840157, 18.841914366552389769835974847342, 19.68274815814992369210620136932, 21.31153801152771582025483783468, 22.15506125205875640583556875560, 22.96149808036860633580863095216, 23.8881608882112306607000137325, 24.28185940160274049452366727587, 25.01364330414502889153667384778

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