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Properties

Label 1-6223-6223.3895-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.930 - 0.367i$
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  + (−0.5 + 0.866i)2-s + (0.698 − 0.715i)3-s + (−0.5 − 0.866i)4-s + (0.623 − 0.781i)5-s + (0.270 + 0.962i)6-s + 8-s + (−0.0249 − 0.999i)9-s + (0.365 + 0.930i)10-s + (−0.318 − 0.947i)11-s + (−0.969 − 0.246i)12-s + (−0.456 + 0.889i)13-s + (−0.124 − 0.992i)15-s + (−0.5 + 0.866i)16-s + (−0.173 − 0.984i)17-s + (0.878 + 0.478i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.698 − 0.715i)3-s + (−0.5 − 0.866i)4-s + (0.623 − 0.781i)5-s + (0.270 + 0.962i)6-s + 8-s + (−0.0249 − 0.999i)9-s + (0.365 + 0.930i)10-s + (−0.318 − 0.947i)11-s + (−0.969 − 0.246i)12-s + (−0.456 + 0.889i)13-s + (−0.124 − 0.992i)15-s + (−0.5 + 0.866i)16-s + (−0.173 − 0.984i)17-s + (0.878 + 0.478i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.930 - 0.367i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (3895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.930 - 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2016045962 - 1.058873106i\)
\(L(\frac12)\) \(\approx\) \(0.2016045962 - 1.058873106i\)
\(L(1)\) \(\approx\) \(0.9322004889 - 0.2423943803i\)
\(L(1)\) \(\approx\) \(0.9322004889 - 0.2423943803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.698 - 0.715i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (-0.318 - 0.947i)T \)
13 \( 1 + (-0.456 + 0.889i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.995 - 0.0995i)T \)
29 \( 1 + (0.969 - 0.246i)T \)
31 \( 1 + (0.318 - 0.947i)T \)
37 \( 1 + (-0.853 + 0.521i)T \)
41 \( 1 + (-0.878 - 0.478i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.969 - 0.246i)T \)
59 \( 1 + (-0.998 - 0.0498i)T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (-0.456 + 0.889i)T \)
71 \( 1 + (0.995 + 0.0995i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.969 - 0.246i)T \)
83 \( 1 + (-0.0249 - 0.999i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + (-0.124 - 0.992i)T \)
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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

−17.83317087451789036700727121568, −17.607710748452687392884210937015, −16.84533060438819416262057011221, −15.81795226468107560952791136388, −15.37554338168399924274312885460, −14.620664229070543807546880313318, −13.94064534142649710560058888571, −13.41974549152357251574068764312, −12.6420078458229797224394841055, −12.043089101954451874523218209365, −10.92024371949619203357259221647, −10.60581106192225746301109888114, −9.90765591420304160858009243175, −9.70275399955623437045176737018, −8.74133162466605992298061946630, −8.14075175875374717360438454189, −7.42848863390725731076603815065, −6.76998167979459113086206600255, −5.55001459922863608135262975090, −4.84749067837743376709215616721, −4.140697722529111440666234136266, −3.22875178550399653925643822968, −2.79102612973894974826652411607, −2.12401004813678849760328769401, −1.40786254538626586894902065664, 0.28228584319793960835381435085, 1.106187156959804617907091886821, 1.90766572475519284653606202685, 2.56572156246097793802177682228, 3.76271850565812874697977482361, 4.54714147600783140548453410856, 5.46764156516605146522877719794, 5.9387652710471051895565785972, 6.73211700342901520700023574938, 7.33443668349275611888177513504, 8.152385115359811708389543127581, 8.60773601961251632915220121267, 9.12958728990568126304403812322, 9.9019402158726563255631572096, 10.306536959099283759329864975274, 11.807375858960048081126428339729, 11.97424070308061045330814653939, 13.17480477286561638054211957630, 13.68513995715939024682561574774, 14.00134789904389965509534891381, 14.55736529331004392207560673504, 15.68538650689726305307026281979, 15.9758885827788764165131095888, 16.87605825446004110694553097299, 17.20911205803925378607642896329

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