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Properties

Label 1-455-455.412-r0-0-0
Degree $1$
Conductor $455$
Sign $-0.235 - 0.971i$
Analytic cond. $2.11301$
Root an. cond. $2.11301$
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)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)6-s − i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·12-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s − i·18-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)6-s − i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s − i·12-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s − i·18-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(2.11301\)
Root analytic conductor: \(2.11301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 455,\ (0:\ ),\ -0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.781042133 - 2.265329845i\)
\(L(\frac12)\) \(\approx\) \(1.781042133 - 2.265329845i\)
\(L(1)\) \(\approx\) \(1.781429227 - 1.185392844i\)
\(L(1)\) \(\approx\) \(1.781429227 - 1.185392844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−24.18365764765016023036662739684, −23.39384438807330248587885887172, −22.42263281793434528820245853964, −21.70685366970613511461315220888, −20.77072839832779533794233252696, −20.35088011042320849739994408818, −19.292948791013750936171227426969, −18.10202861560843141231129878718, −17.04607236627829617177198986243, −16.04354867708993677406944109700, −15.462391923408414086088285514, −14.613613832938397194701002529627, −13.94278519974917100951129876694, −13.011484549515517177266416373707, −12.235557990601397869969072656706, −10.98242500696923627699575214007, −9.9622037383572165327382018845, −8.91150510034420118962717930804, −7.83713766092552360817299860429, −7.2528634767517908918150324160, −5.90878561386029188354063196008, −4.77257748509098395433228172161, −4.12821071083290114099428961143, −2.90872840840612054689580274236, −2.143008234248460055626422373912, 1.19162626667797002511725839044, 2.26260811543794111301954828754, 3.31205798566715392232759730027, 4.0100747655873168300413848841, 5.50734286783121559823672101451, 6.29203016635377351284828302489, 7.52013053454201026417994089830, 8.3937955779006351737663410966, 9.62017964714548307576007995157, 10.48823734935058639400807929595, 11.57005913968123563588681279868, 12.61737082329277804801751429627, 13.08795606439791376807937793882, 14.26306810125189309122234130882, 14.50197657906226090665775745395, 15.69397899913618396676789588202, 16.519424844000536242281547507243, 18.144048099626498735151812246655, 18.79284124882983243073478318462, 19.60897039940174078189612669300, 20.27112884439052948519721504990, 21.36376317434066249636843791986, 21.58600761180417244245161761889, 23.067816741707061815064597910067, 23.68919326118359130726866675255

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