↑

Properties

Label 1-3009-3009.41-r0-0-0
Degree $1$
Conductor $3009$
Sign $-0.863 - 0.505i$
Analytic cond. $13.9737$
Root an. cond. $13.9737$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.667 + 0.744i)2-s + (−0.108 − 0.994i)4-s + (−0.753 + 0.657i)5-s + (0.835 − 0.549i)7-s + (0.812 + 0.583i)8-s + (0.0135 − 0.999i)10-s + (−0.572 + 0.820i)11-s + (−0.762 − 0.647i)13-s + (−0.148 + 0.988i)14-s + (−0.976 + 0.214i)16-s + (0.293 − 0.955i)19-s + (0.735 + 0.677i)20-s + (−0.228 − 0.973i)22-s + (−0.913 + 0.407i)23-s + (0.135 − 0.990i)25-s + (0.990 − 0.135i)26-s + ⋯
L(s)  = 1  + (−0.667 + 0.744i)2-s + (−0.108 − 0.994i)4-s + (−0.753 + 0.657i)5-s + (0.835 − 0.549i)7-s + (0.812 + 0.583i)8-s + (0.0135 − 0.999i)10-s + (−0.572 + 0.820i)11-s + (−0.762 − 0.647i)13-s + (−0.148 + 0.988i)14-s + (−0.976 + 0.214i)16-s + (0.293 − 0.955i)19-s + (0.735 + 0.677i)20-s + (−0.228 − 0.973i)22-s + (−0.913 + 0.407i)23-s + (0.135 − 0.990i)25-s + (0.990 − 0.135i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3009\)    =    \(3 \cdot 17 \cdot 59\)
Sign: $-0.863 - 0.505i$
Analytic conductor: \(13.9737\)
Root analytic conductor: \(13.9737\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3009} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3009,\ (0:\ ),\ -0.863 - 0.505i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04022816238 + 0.1484110588i\)
\(L(\frac12)\) \(\approx\) \(-0.04022816238 + 0.1484110588i\)
\(L(1)\) \(\approx\) \(0.5510408220 + 0.2084385580i\)
\(L(1)\) \(\approx\) \(0.5510408220 + 0.2084385580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.667 + 0.744i)T \)
5 \( 1 + (-0.753 + 0.657i)T \)
7 \( 1 + (0.835 - 0.549i)T \)
11 \( 1 + (-0.572 + 0.820i)T \)
13 \( 1 + (-0.762 - 0.647i)T \)
19 \( 1 + (0.293 - 0.955i)T \)
23 \( 1 + (-0.913 + 0.407i)T \)
29 \( 1 + (0.332 + 0.943i)T \)
31 \( 1 + (0.995 + 0.0946i)T \)
37 \( 1 + (0.973 - 0.228i)T \)
41 \( 1 + (-0.933 + 0.357i)T \)
43 \( 1 + (-0.538 + 0.842i)T \)
47 \( 1 + (0.319 - 0.947i)T \)
53 \( 1 + (-0.918 + 0.395i)T \)
61 \( 1 + (-0.332 + 0.943i)T \)
67 \( 1 + (0.161 + 0.986i)T \)
71 \( 1 + (-0.0676 - 0.997i)T \)
73 \( 1 + (0.988 + 0.148i)T \)
79 \( 1 + (-0.0406 - 0.999i)T \)
83 \( 1 + (-0.870 + 0.492i)T \)
89 \( 1 + (-0.998 - 0.0541i)T \)
97 \( 1 + (-0.594 + 0.804i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.7361407687711752284352577908, −18.21807818244652543044188769967, −17.14030677718817917107023924943, −16.8294456306629857010756306273, −15.91037063107621556358839419042, −15.45520315286889005191672421161, −14.286100083174816763093200987923, −13.69279164362456103907509523426, −12.65350720724777693025142096227, −12.08331610521078270717049803603, −11.631579766079260745901386532736, −10.99441533754709636596830427766, −10.04015972344694504893822302222, −9.40063151581333231598392417788, −8.37665941291782596080748752306, −8.181745450848518710691331179484, −7.55651716009090961036201297787, −6.36392621848691866861048732303, −5.27480634621874717125448463261, −4.53706840731796385705026492332, −3.86628072416114364645732720336, −2.84938926689058385677561753553, −2.06004588233783224051263651020, −1.17953673682034988941387714976, −0.06886958723457176327240546604, 1.10076322191955579079190975905, 2.19336738099319573390346430239, 3.068149708367373649725264966123, 4.42194641530601345100778748022, 4.785294563504447430905929909300, 5.702108291887872757908535097795, 6.89001894627544810879858427808, 7.19573062824468123965945497361, 7.99329163430157208970820952956, 8.31534074488518482683134240406, 9.600333801671434380773879393311, 10.17078211494916220693118049844, 10.79237342874876007468893031342, 11.4920827597263414706885272830, 12.2753951873289964550927859008, 13.40311774737790119174622809528, 14.081742082399773521199650903539, 14.94172195612253566480511263419, 15.14864008160778399435055021397, 15.923663319377572817836919000103, 16.69239111955340787497539635799, 17.64127174078658823749452659347, 17.895311460074130164804429428448, 18.498770660497303825629945620144, 19.54412138652655582173752253557

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