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Properties

Label 1-6223-6223.891-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.947 + 0.318i$
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  + (−0.733 + 0.680i)2-s + (−0.733 − 0.680i)3-s + (0.0747 − 0.997i)4-s + (0.826 + 0.563i)5-s + 6-s + (0.623 + 0.781i)8-s + (0.0747 + 0.997i)9-s + (−0.988 + 0.149i)10-s + (0.0747 − 0.997i)11-s + (−0.733 + 0.680i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−0.733 − 0.680i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (−0.733 − 0.680i)3-s + (0.0747 − 0.997i)4-s + (0.826 + 0.563i)5-s + 6-s + (0.623 + 0.781i)8-s + (0.0747 + 0.997i)9-s + (−0.988 + 0.149i)10-s + (0.0747 − 0.997i)11-s + (−0.733 + 0.680i)12-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.988 − 0.149i)16-s + (0.826 − 0.563i)17-s + (−0.733 − 0.680i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (891, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.947 + 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02813737066 + 0.1720873786i\)
\(L(\frac12)\) \(\approx\) \(0.02813737066 + 0.1720873786i\)
\(L(1)\) \(\approx\) \(0.5911012561 + 0.06499996978i\)
\(L(1)\) \(\approx\) \(0.5911012561 + 0.06499996978i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.733 + 0.680i)T \)
3 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.733 - 0.680i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.733 - 0.680i)T \)
79 \( 1 + (0.365 - 0.930i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−17.34373070400401912609896101957, −16.93775301647317757663451996836, −16.403903381228673699431655097207, −15.526711930895373811584164630453, −14.92412563291818158997361817115, −14.10400064740060392530798874786, −12.87123052794799710553337605674, −12.70168146193177958469348149464, −12.172965505767704207442144842714, −11.25513403870684979569864548157, −10.55737052046559911405008425761, −10.0526174520476128824498141390, −9.66487553940090944050277648420, −8.95530096200182725896040556165, −8.22869814376453760184364566429, −7.39219613344413384948423935146, −6.604926952717155754789234314138, −5.689660127831809053034250747639, −5.162538931113530177221827237318, −4.36648142698844697423565079997, −3.59239033335099843053657088866, −2.85154454746075501891660399318, −1.675529687484409320260165738149, −1.334538640188508038740801936208, −0.06844387305661643051575607430, 0.970161742269896888334436501532, 1.77763915695471891054942970230, 2.39413928640575868619909807814, 3.4370653559321934670280240209, 4.86279631658911307037245826950, 5.303568698518774245372852952191, 6.00313030119604002177632337001, 6.67185900232424835619427763445, 7.02973103248647053274453866832, 7.72326503575486852069470864501, 8.70696213258983434193194198450, 9.21175454328129494838402190763, 9.9800258876568192474476812988, 10.75337649454665208609073472613, 11.20809915164956372611811659129, 11.73627167448846328650467457838, 12.93389011871885132678782358162, 13.42365382298797950005753044041, 14.19507115225278216349634841678, 14.5196105365608302247241575314, 15.46503021535445140251045144952, 16.37800169960932340631374683207, 16.748106229728276145667973622778, 17.23374622916936122861069283861, 17.88714632081489321033135198334

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