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Properties

Label 1-420-420.107-r1-0-0
Degree $1$
Conductor $420$
Sign $0.578 - 0.815i$
Analytic cond. $45.1352$
Root an. cond. $45.1352$
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  + (−0.5 + 0.866i)11-s − i·13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + 29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)37-s − 41-s − i·43-s + (0.866 − 0.5i)47-s + (−0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)11-s − i·13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + 29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)37-s − 41-s − i·43-s + (0.866 − 0.5i)47-s + (−0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (−0.5 − 0.866i)61-s + (0.866 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(45.1352\)
Root analytic conductor: \(45.1352\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 420,\ (1:\ ),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.459922083 - 0.7543513700i\)
\(L(\frac12)\) \(\approx\) \(1.459922083 - 0.7543513700i\)
\(L(1)\) \(\approx\) \(1.045608048 - 0.09732616794i\)
\(L(1)\) \(\approx\) \(1.045608048 - 0.09732616794i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−23.9334938065649561375887099948, −23.48286222552092610987267378207, −22.37520239463095997947955220690, −21.426494282551480692479700870113, −20.88547958723925768681303570749, −19.75469567658133601525759646096, −18.797089649814561356641089773047, −18.32078642807974005496532658505, −16.95520043367058573426607873161, −16.368491317507570211916054568346, −15.48425562190542985019700819362, −14.17328767513794663200436729546, −13.84167992176810061519367086375, −12.46496903884925628311647425802, −11.78668077772905774062558886439, −10.66026554010675502445387450670, −9.86373493184672129822757506829, −8.67045770212350719498517033158, −7.92841361529069897905080006462, −6.69107516094919244102366006112, −5.79946003035810532643039255446, −4.658227504014382444246808284496, −3.53525145674408484963749600530, −2.38297658287480656294076215321, −0.996188950512290141695270630557, 0.526838685381652222100906417033, 2.02359897824396613168286740701, 3.13709545177675729799077973991, 4.39333926027289140036825852719, 5.39386348380666874437025799411, 6.43360626213394056195638997274, 7.64250731558098140484033334555, 8.29166802534649320301777329244, 9.69269583530282160160115700013, 10.29015135913559631703221862526, 11.3908266766722709672273248269, 12.48382006697561026416555346439, 13.10192543987073478540632987448, 14.2418467346139678990984936941, 15.22026223910581052787362992589, 15.80180370218556262258854896567, 17.08141831742969032966507397984, 17.724643338498765773502115888610, 18.60766363852978370504019264954, 19.687132620337775046089576236341, 20.341567884440367724973302705501, 21.320215757153344429112565863095, 22.12043423773576471713291382962, 23.184602674686962019996963277032, 23.65827422995753223867971359889

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