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Properties

Label 1-4600-4600.1931-r0-0-0
Degree $1$
Conductor $4600$
Sign $-0.929 - 0.368i$
Analytic cond. $21.3623$
Root an. cond. $21.3623$
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  + (0.309 − 0.951i)3-s + 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (−0.809 − 0.587i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (−0.809 − 0.587i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (1931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3246759314 - 1.702010836i\)
\(L(\frac12)\) \(\approx\) \(0.3246759314 - 1.702010836i\)
\(L(1)\) \(\approx\) \(1.073457023 - 0.6052934367i\)
\(L(1)\) \(\approx\) \(1.073457023 - 0.6052934367i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−18.421642372212644163109048449540, −17.74151334930505014986114697308, −17.00546150245436498928695623356, −16.671583447322224653121233868531, −15.609434203974044527014393750916, −15.04051253260434336099160696273, −14.763133369617808848406962430865, −13.88330873124138714203903620759, −13.34608658374331006530621110706, −12.215718932096972949621010132635, −11.73126499989016398457856995326, −10.79678713972090966667579584186, −10.46344614012608068961662927081, −9.70025856071307381426757546675, −8.78131664188412177481623596307, −8.36658391002362025801143391746, −7.76179843689039641745292829781, −6.65113641500047695235553749378, −5.864839185876042171158400619504, −5.142943495371940674098555156822, −4.35021232728814483684412600877, −3.835506330495118567849269147421, −3.09352777776290829674933838069, −1.880197267142489503144718203391, −1.44135777338784647250136721946, 0.41106642680128340481114749178, 1.45882801905262441540463361643, 1.890973348634173461238009355709, 2.91874623597853383475426370420, 3.68728800873879514327916125405, 4.58429037007480099892476200065, 5.418340605410148153565558653232, 6.25104570735537055608905747016, 6.91916949759320037733330442973, 7.46587126093402221440931577930, 8.41546808992287686563103137463, 8.85628682999939414949167068728, 9.36189019392420618586366075585, 10.722081210461514693444362382185, 11.37729838092373073684773212203, 11.65460815580777702914853679103, 12.49920277225532519516717999517, 13.52203129446148114392669332329, 13.67883912337481792032208197811, 14.49327643133122778877909344177, 14.97871947266778867346948115929, 15.95601616193219239130647055625, 16.69288907647790376144532568869, 17.45185263988355894311275250199, 17.90615237600800090883484399094

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