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Properties

Label 1-335-335.58-r0-0-0
Degree $1$
Conductor $335$
Sign $0.0387 + 0.999i$
Analytic cond. $1.55573$
Root an. cond. $1.55573$
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  + (−0.909 + 0.415i)2-s + (0.540 − 0.841i)3-s + (0.654 − 0.755i)4-s + (−0.142 + 0.989i)6-s + (−0.909 + 0.415i)7-s + (−0.281 + 0.959i)8-s + (−0.415 − 0.909i)9-s + (0.142 + 0.989i)11-s + (−0.281 − 0.959i)12-s + (0.281 + 0.959i)13-s + (0.654 − 0.755i)14-s + (−0.142 − 0.989i)16-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)18-s + (−0.415 + 0.909i)19-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)2-s + (0.540 − 0.841i)3-s + (0.654 − 0.755i)4-s + (−0.142 + 0.989i)6-s + (−0.909 + 0.415i)7-s + (−0.281 + 0.959i)8-s + (−0.415 − 0.909i)9-s + (0.142 + 0.989i)11-s + (−0.281 − 0.959i)12-s + (0.281 + 0.959i)13-s + (0.654 − 0.755i)14-s + (−0.142 − 0.989i)16-s + (−0.755 + 0.654i)17-s + (0.755 + 0.654i)18-s + (−0.415 + 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0387 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0387 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(335\)    =    \(5 \cdot 67\)
Sign: $0.0387 + 0.999i$
Analytic conductor: \(1.55573\)
Root analytic conductor: \(1.55573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{335} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 335,\ (0:\ ),\ 0.0387 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4479214853 + 0.4308785707i\)
\(L(\frac12)\) \(\approx\) \(0.4479214853 + 0.4308785707i\)
\(L(1)\) \(\approx\) \(0.6694248744 + 0.1167479572i\)
\(L(1)\) \(\approx\) \(0.6694248744 + 0.1167479572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.909 + 0.415i)T \)
3 \( 1 + (0.540 - 0.841i)T \)
7 \( 1 + (-0.909 + 0.415i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (-0.755 + 0.654i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 - T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.540 - 0.841i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + iT \)
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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

−25.12831324629562820619542340098, −24.17848587604124497825054514392, −22.4287385297250273575657092641, −22.15548073807067906531007717300, −20.88257386897212332586502013912, −20.279064787351413589159234131617, −19.50566772070972151405374406524, −18.78144720471323155617186915754, −17.5393401006206875286727334886, −16.603945390374144083820272274299, −15.9360156447820430156733701699, −15.197761037878284138251183445364, −13.67417887085664980740641185648, −13.02332844586869805063749944439, −11.5407676268893859133481860961, −10.67119698870948562969304794873, −9.99944905107988323256238616886, −8.98690128191592089169068729799, −8.366637030084716916273150939072, −7.14149133359947755879741449769, −5.958893704991288532405660669663, −4.27323530244349684403392177350, −3.25201310026016327852233303538, −2.50722499810762881702726447763, −0.47306605943173102933087393776, 1.5928075339063779653457995461, 2.36934559773858910608242553407, 3.92050823405308195024204371525, 5.8549650078994114235267176558, 6.58709805691879231999840097395, 7.39211586110171612699986601098, 8.48365592061577613526854594643, 9.27357839571806146944859564021, 10.050315623763282123482534438539, 11.5091562470502571827028670814, 12.38911000906032249394340392631, 13.40328590874236528556732527085, 14.555551669291056579963757318460, 15.282088916942608755187586421154, 16.2899746061176011576881211607, 17.315361238334131656830149397052, 18.14320717447185032930483187009, 18.968344827868107487650188188853, 19.57838438645203487498506471913, 20.32426842370861033109776865649, 21.47763760407625866737743026043, 22.97829352716913656582452283414, 23.57021142187638028285244935815, 24.69140500024575134053302533507, 25.20257255560329408750664241063

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