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Properties

Label 1-336-336.5-r0-0-0
Degree $1$
Conductor $336$
Sign $0.999 + 0.00414i$
Analytic cond. $1.56037$
Root an. cond. $1.56037$
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  + (0.866 − 0.5i)5-s + (0.866 + 0.5i)11-s − i·13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)37-s − 41-s + i·43-s + (−0.5 − 0.866i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (0.866 + 0.5i)11-s − i·13-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)37-s − 41-s + i·43-s + (−0.5 − 0.866i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.999 + 0.00414i$
Analytic conductor: \(1.56037\)
Root analytic conductor: \(1.56037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 336,\ (0:\ ),\ 0.999 + 0.00414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.550114323 + 0.003215942838i\)
\(L(\frac12)\) \(\approx\) \(1.550114323 + 0.003215942838i\)
\(L(1)\) \(\approx\) \(1.270923685 + 0.01606433274i\)
\(L(1)\) \(\approx\) \(1.270923685 + 0.01606433274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−25.01972345663896170114360795028, −24.33637650648852108867986089659, −22.92671486927418558422764730405, −22.341345049608599103930938847652, −21.56241230611562154640768695112, −20.54047117684224459207084253303, −19.694526099740396128132934507681, −18.57497097470191245642001934004, −17.82524602701323336096951205328, −17.06990641118137962880731255148, −15.95249795321844203355753038118, −14.97891749935174626376263808568, −13.90396172758531655606844384729, −13.46954737716334173347765136815, −12.07557445077881039435992169553, −11.20536920616035314691521271663, −10.0760301591681752976267143903, −9.41355543648612060642623197116, −8.18747081244723565231034205917, −7.01615451187565750672830994839, −6.04682835956369089270146017268, −5.18111484624900795238360327564, −3.62612102910112236847155476746, −2.633862588005405975231219851018, −1.24259500811035937656889056821, 1.35478633183808674151531925888, 2.33990608662617695924676572527, 3.99678041241963724143004704019, 4.91028887330997328750207969819, 6.17255552374556587261379096120, 6.9052117296251377843365246835, 8.40380659355716354506443079555, 9.27150523645273409071251782942, 9.99625918958042123834724628606, 11.26934746463824206799241677859, 12.24259983656474015145522358024, 13.1715159644852011142123208341, 14.086750735406522991726120797700, 14.86975261641528933724575728831, 16.19498831235605962585701196198, 16.92342465766812045422914636642, 17.72406997680473266963385660476, 18.628996422972077924776429480183, 19.868566913876591882558595438761, 20.41304580832105011074377472883, 21.68622362800615379540028632635, 21.99632702082445174373775006814, 23.25162468933892251113574043163, 24.35366018289484500046830856629, 24.76859184021424776694730053910

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