Properties

Label 1-379-379.378-r1-0-0
Degree $1$
Conductor $379$
Sign $1$
Analytic cond. $40.7292$
Root an. cond. $40.7292$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(379\)
Sign: $1$
Analytic conductor: \(40.7292\)
Root analytic conductor: \(40.7292\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{379} (378, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 379,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5682661129\)
\(L(\frac12)\) \(\approx\) \(0.5682661129\)
\(L(1)\) \(\approx\) \(0.4841183254\)
\(L(1)\) \(\approx\) \(0.4841183254\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad379 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−24.55566157767619259372651007915, −23.61092924749741532818328746887, −22.3820308354628642211570022891, −21.816973517670714165495845616781, −20.82048004907735304596676239523, −19.85105634669550411838001887778, −18.719580017804240096607874347732, −18.13195322936877710140654937286, −17.331023218310068771611457349939, −16.577439136434493840316855379351, −15.90283561621656083121415190093, −14.869693832128888825189727313498, −13.15171858562111825928069038689, −12.71877323044733548781095949789, −11.39985909082935667398890028901, −10.59121063111293054064315182790, −9.72327289118219699625228889941, −9.25529546686854271211317295340, −7.53807341853369624573845275705, −6.79258886478060940585175967493, −5.872971104654691891251309608280, −5.02290360857028008474067564375, −3.02324158095572599282478064072, −1.94740423360350370719059513847, −0.50538459045467819629987870038, 0.50538459045467819629987870038, 1.94740423360350370719059513847, 3.02324158095572599282478064072, 5.02290360857028008474067564375, 5.872971104654691891251309608280, 6.79258886478060940585175967493, 7.53807341853369624573845275705, 9.25529546686854271211317295340, 9.72327289118219699625228889941, 10.59121063111293054064315182790, 11.39985909082935667398890028901, 12.71877323044733548781095949789, 13.15171858562111825928069038689, 14.869693832128888825189727313498, 15.90283561621656083121415190093, 16.577439136434493840316855379351, 17.331023218310068771611457349939, 18.13195322936877710140654937286, 18.719580017804240096607874347732, 19.85105634669550411838001887778, 20.82048004907735304596676239523, 21.816973517670714165495845616781, 22.3820308354628642211570022891, 23.61092924749741532818328746887, 24.55566157767619259372651007915

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