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Properties

Label 1-6223-6223.1056-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.997 - 0.0758i$
Analytic cond. $28.8994$
Root an. cond. $28.8994$
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  + 2-s + (−0.988 + 0.149i)3-s + 4-s + (−0.222 + 0.974i)5-s + (−0.988 + 0.149i)6-s + 8-s + (0.955 − 0.294i)9-s + (−0.222 + 0.974i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.826 + 0.563i)13-s + (0.0747 − 0.997i)15-s + 16-s + (0.5 + 0.866i)17-s + (0.955 − 0.294i)18-s − 19-s + ⋯
L(s)  = 1  + 2-s + (−0.988 + 0.149i)3-s + 4-s + (−0.222 + 0.974i)5-s + (−0.988 + 0.149i)6-s + 8-s + (0.955 − 0.294i)9-s + (−0.222 + 0.974i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.826 + 0.563i)13-s + (0.0747 − 0.997i)15-s + 16-s + (0.5 + 0.866i)17-s + (0.955 − 0.294i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.997 - 0.0758i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1056, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.997 - 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04892231235 + 1.288702356i\)
\(L(\frac12)\) \(\approx\) \(-0.04892231235 + 1.288702356i\)
\(L(1)\) \(\approx\) \(1.111564728 + 0.5024210847i\)
\(L(1)\) \(\approx\) \(1.111564728 + 0.5024210847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + (0.733 + 0.680i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (-0.955 + 0.294i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (0.826 + 0.563i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (-0.826 + 0.563i)T \)
71 \( 1 + (0.365 + 0.930i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.988 + 0.149i)T \)
83 \( 1 + (0.955 - 0.294i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.0747 - 0.997i)T \)
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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

−17.1812037268189722545877957213, −16.51897213970369769578456111692, −16.014818487036146625785536941970, −15.48941785610606610203499817322, −14.828988581961955920519490432312, −13.58765855191772607469952207533, −13.48196164854930380188616634892, −12.613409390403771172283547965911, −12.147958249264371386959751291897, −11.65269191023913783919273520775, −10.95804318613914048585808381561, −10.186560553169784466021206051278, −9.654591562353019218581287505813, −8.35648273811351582674095189447, −7.783699336587750439015031555428, −7.19648839030511281109069846385, −6.19119377610234498614450893162, −5.72611724536976876682705007605, −4.99359177835110059951510960241, −4.69166603444728635887866974840, −3.82382428986803019337486228926, −2.880315031974323399759661625857, −2.06080708418857330699282969798, −1.069853036150330217039765340801, −0.26140630711007914581292288556, 1.31937845379893374145842013598, 2.30454912601505478002285268389, 2.781025105711198004459411762383, 3.92596623880588932662138393716, 4.404612763060381229700940103725, 5.00429995898823336421689083204, 5.912589273413215363282686303241, 6.4576175604812579921169536455, 6.98071763085245490562083427078, 7.61493181963156886552111948440, 8.45003294013577674666521736556, 10.01613623897710207044924482777, 10.22970192436823084425192294698, 10.719825336439747175140007727340, 11.63638808859526746448201661282, 12.02705087728508181712127262671, 12.67206216278275215591958224499, 13.2418302265751077355640914706, 14.24018822267922022226193636725, 14.87826398779375556783387804180, 15.1156368981921072664132209488, 16.01343848049778688368786876455, 16.512875213291820041803984898751, 17.24394791133622755675911193296, 17.9035743822522156346768495932

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