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Properties

Label 1-3724-3724.3259-r0-0-0
Degree $1$
Conductor $3724$
Sign $-0.547 + 0.837i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (0.542 + 0.840i)3-s + (0.542 + 0.840i)5-s + (−0.411 + 0.911i)9-s + (0.900 + 0.433i)11-s + (−0.270 + 0.962i)13-s + (−0.411 + 0.911i)15-s + (0.980 − 0.198i)17-s + (0.661 − 0.749i)23-s + (−0.411 + 0.911i)25-s + (−0.988 + 0.149i)27-s + (0.853 − 0.521i)29-s + (−0.5 + 0.866i)31-s + (0.124 + 0.992i)33-s + (−0.365 − 0.930i)37-s + (−0.955 + 0.294i)39-s + ⋯
L(s)  = 1  + (0.542 + 0.840i)3-s + (0.542 + 0.840i)5-s + (−0.411 + 0.911i)9-s + (0.900 + 0.433i)11-s + (−0.270 + 0.962i)13-s + (−0.411 + 0.911i)15-s + (0.980 − 0.198i)17-s + (0.661 − 0.749i)23-s + (−0.411 + 0.911i)25-s + (−0.988 + 0.149i)27-s + (0.853 − 0.521i)29-s + (−0.5 + 0.866i)31-s + (0.124 + 0.992i)33-s + (−0.365 − 0.930i)37-s + (−0.955 + 0.294i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.547 + 0.837i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (3259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ -0.547 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299302980 + 2.401213657i\)
\(L(\frac12)\) \(\approx\) \(1.299302980 + 2.401213657i\)
\(L(1)\) \(\approx\) \(1.302581443 + 0.8316174375i\)
\(L(1)\) \(\approx\) \(1.302581443 + 0.8316174375i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.542 + 0.840i)T \)
5 \( 1 + (0.542 + 0.840i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.270 + 0.962i)T \)
17 \( 1 + (0.980 - 0.198i)T \)
23 \( 1 + (0.661 - 0.749i)T \)
29 \( 1 + (0.853 - 0.521i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.365 - 0.930i)T \)
41 \( 1 + (0.998 - 0.0498i)T \)
43 \( 1 + (0.797 - 0.603i)T \)
47 \( 1 + (0.969 - 0.246i)T \)
53 \( 1 + (0.318 + 0.947i)T \)
59 \( 1 + (-0.797 + 0.603i)T \)
61 \( 1 + (0.878 + 0.478i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.0249 + 0.999i)T \)
73 \( 1 + (0.698 - 0.715i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (-0.995 + 0.0995i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.401004618147736113315070828015, −17.60647832857850880828129123993, −17.183002628962409336675840061342, −16.53727321851411993430160902823, −15.58746969420637073788004022074, −14.79401986285373792381533974770, −14.1349003119897794608327015392, −13.60694316572436302791770922468, −12.77288497800189287035246099333, −12.44664031474646646791081710294, −11.68978069445453183470329833925, −10.78468139500003292594732383700, −9.67172815094443772039901083571, −9.34656504826843688782126045863, −8.45164701378884881878931291270, −7.96457737282502864786377567108, −7.17060280248892301607295325233, −6.236595696656914419401655276879, −5.707465246908983308536353040691, −4.906793319974757677361883047329, −3.77384609371397733650326218612, −3.107190665855624645768182301063, −2.20106708171225964930930538329, −1.230657979221731964101034966548, −0.8156691206403897143230509160, 1.267566161496756478775233337396, 2.299775834170406803237850426279, 2.79859797103293020928982430089, 3.810054063275342764910102677788, 4.30508983140830714973065130324, 5.28981895150790508583790237407, 6.02184806417619810900068546126, 7.01029795366393176488016228140, 7.39523881386904167586850729616, 8.61468647421668232838572987891, 9.18921013210682101356276934997, 9.754239544614397760752094316681, 10.45530984623250828300722409168, 11.01701919389571463925757070350, 11.89915134693511201599262952975, 12.595798554380552859905215083481, 13.80207140649157675459075949174, 14.20213050961285442064737185669, 14.574942408690612516801055994607, 15.30971842333719431698503353798, 16.116165673805738044958774300317, 16.82338354691621317900974593577, 17.34051877185396768603962705970, 18.20565117274193876013664535500, 19.129280002503108047074902540513

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