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Properties

Label 1-2580-2580.1883-r0-0-0
Degree $1$
Conductor $2580$
Sign $-0.983 - 0.178i$
Analytic cond. $11.9814$
Root an. cond. $11.9814$
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  + (−0.866 + 0.5i)7-s + (−0.900 + 0.433i)11-s + (0.149 − 0.988i)13-s + (0.930 + 0.365i)17-s + (−0.826 + 0.563i)19-s + (0.997 + 0.0747i)23-s + (−0.955 + 0.294i)29-s + (0.733 + 0.680i)31-s + (−0.866 − 0.5i)37-s + (0.222 − 0.974i)41-s + (−0.433 + 0.900i)47-s + (0.5 − 0.866i)49-s + (0.149 + 0.988i)53-s + (−0.623 − 0.781i)59-s + (0.733 − 0.680i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.900 + 0.433i)11-s + (0.149 − 0.988i)13-s + (0.930 + 0.365i)17-s + (−0.826 + 0.563i)19-s + (0.997 + 0.0747i)23-s + (−0.955 + 0.294i)29-s + (0.733 + 0.680i)31-s + (−0.866 − 0.5i)37-s + (0.222 − 0.974i)41-s + (−0.433 + 0.900i)47-s + (0.5 − 0.866i)49-s + (0.149 + 0.988i)53-s + (−0.623 − 0.781i)59-s + (0.733 − 0.680i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2580\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 43\)
Sign: $-0.983 - 0.178i$
Analytic conductor: \(11.9814\)
Root analytic conductor: \(11.9814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2580} (1883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2580,\ (0:\ ),\ -0.983 - 0.178i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008447270531 + 0.09369004944i\)
\(L(\frac12)\) \(\approx\) \(0.008447270531 + 0.09369004944i\)
\(L(1)\) \(\approx\) \(0.7706364360 + 0.07267624747i\)
\(L(1)\) \(\approx\) \(0.7706364360 + 0.07267624747i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.149 - 0.988i)T \)
17 \( 1 + (0.930 + 0.365i)T \)
19 \( 1 + (-0.826 + 0.563i)T \)
23 \( 1 + (0.997 + 0.0747i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (0.733 + 0.680i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.149 + 0.988i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.563 + 0.826i)T \)
71 \( 1 + (-0.0747 - 0.997i)T \)
73 \( 1 + (-0.149 + 0.988i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.294 + 0.955i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
97 \( 1 + (0.433 + 0.900i)T \)
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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

−18.91298039986581071842760064919, −18.62616161180144531934727121918, −17.47815230265067447948778463953, −16.68439981469038104817646974962, −16.39531280983599343759307606510, −15.47704026142835205295242364014, −14.81746495880059013872406689341, −13.83048608186206290014340140553, −13.30225480081826741444281805494, −12.756500242579147197053493525754, −11.734010928028548574595277649842, −11.09260505125823126005235472222, −10.26091955482512944563420499684, −9.653484054652960690437507740568, −8.85344753696512657235060517133, −8.035435370507489424406424869412, −7.13929621947014802898443691896, −6.57733854703419419655872660400, −5.69818307378367366007149066211, −4.83869004093255137541666877667, −3.957685159201833619513447802964, −3.13622613016384225621379115484, −2.39523564717734727344656373993, −1.16909130168383579629694344786, −0.03163743353593348684555425888, 1.33669971912921767845631469735, 2.46049901488841468153121867823, 3.14359195541078934514363275122, 3.900282893071813016073516331489, 5.16263190773936230322856480941, 5.59719776046136836009900863658, 6.46655075024172990534734538433, 7.35018574520095893548098841481, 8.08373655943364193845964212480, 8.8417946045342838538617895538, 9.70254217903453442840150471436, 10.39773352516111461434305367966, 10.892486895361584566670066153, 12.13555106016137928249456646246, 12.728831469510804773831562126051, 13.00305085609019577913691577491, 14.10698892415765005501173777225, 14.92631167052781437438632261020, 15.54205959389616438434332349102, 16.06358766281501300149716362675, 17.01222588245626133033826692325, 17.56084861856260753623080417606, 18.565664145850549402114588183231, 18.9353531500107116095410278248, 19.645468283323712448475582938449

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