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Properties

Label 1-4600-4600.3261-r0-0-0
Degree $1$
Conductor $4600$
Sign $-0.368 + 0.929i$
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  + (0.466 − 0.884i)3-s + (−0.654 + 0.755i)7-s + (−0.564 − 0.825i)9-s + (0.254 + 0.967i)11-s + (−0.516 + 0.856i)13-s + (0.198 + 0.980i)17-s + (0.870 + 0.491i)19-s + (0.362 + 0.931i)21-s + (−0.993 + 0.113i)27-s + (0.870 − 0.491i)29-s + (−0.466 − 0.884i)31-s + (0.974 + 0.226i)33-s + (0.564 + 0.825i)37-s + (0.516 + 0.856i)39-s + (−0.0285 − 0.999i)41-s + ⋯
L(s)  = 1  + (0.466 − 0.884i)3-s + (−0.654 + 0.755i)7-s + (−0.564 − 0.825i)9-s + (0.254 + 0.967i)11-s + (−0.516 + 0.856i)13-s + (0.198 + 0.980i)17-s + (0.870 + 0.491i)19-s + (0.362 + 0.931i)21-s + (−0.993 + 0.113i)27-s + (0.870 − 0.491i)29-s + (−0.466 − 0.884i)31-s + (0.974 + 0.226i)33-s + (0.564 + 0.825i)37-s + (0.516 + 0.856i)39-s + (−0.0285 − 0.999i)41-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (3261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ -0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5550889213 + 0.8169843111i\)
\(L(\frac12)\) \(\approx\) \(0.5550889213 + 0.8169843111i\)
\(L(1)\) \(\approx\) \(1.010305571 + 0.01453858434i\)
\(L(1)\) \(\approx\) \(1.010305571 + 0.01453858434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.466 - 0.884i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (0.254 + 0.967i)T \)
13 \( 1 + (-0.516 + 0.856i)T \)
17 \( 1 + (0.198 + 0.980i)T \)
19 \( 1 + (0.870 + 0.491i)T \)
29 \( 1 + (0.870 - 0.491i)T \)
31 \( 1 + (-0.466 - 0.884i)T \)
37 \( 1 + (0.564 + 0.825i)T \)
41 \( 1 + (-0.0285 - 0.999i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.974 + 0.226i)T \)
59 \( 1 + (-0.516 + 0.856i)T \)
61 \( 1 + (-0.897 - 0.441i)T \)
67 \( 1 + (0.998 - 0.0570i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (0.993 - 0.113i)T \)
79 \( 1 + (0.0855 + 0.996i)T \)
83 \( 1 + (-0.610 + 0.791i)T \)
89 \( 1 + (-0.985 - 0.170i)T \)
97 \( 1 + (0.610 + 0.791i)T \)
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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

−17.92938163434247112646075350174, −17.18637226000325811816839194217, −16.46946476888444313158838539938, −15.97340349720440367229304755782, −15.57923972068353517683236669815, −14.31844253085715173263282495215, −14.25901000027603962729564596654, −13.412227784693250411022023687780, −12.76045865748372071904315848168, −11.77986372623536688205796502006, −11.00492532694117781836265518479, −10.51055278592544148896062193061, −9.68057947112211949592253876357, −9.33585257989204172547643841053, −8.44783379427461535925993901985, −7.71115314540959427824156296845, −7.04921461927111412325678574778, −6.114903082004382996292041263664, −5.23314589461380081612854735282, −4.72672495446973606416128961792, −3.7101020685142844663282718321, −3.10112946185352456825610397323, −2.73519481944220160650971392233, −1.22936617938220234958940513583, −0.25316239704726489200036256427, 1.252831555593744971840177671849, 1.97013959312362304870047433458, 2.61658033305745043321232258325, 3.45845344526942105692389659362, 4.26664796312345003580821707651, 5.26357432766809650371843008222, 6.2117835254147697452656374434, 6.5483636593060318415118737316, 7.43935564853232675612957048817, 8.019456744224121405735092918683, 8.82029202242521720007044586977, 9.656424622670550450406509907615, 9.8070265098903446145172528249, 11.16150764323044003491989540555, 12.00145308273759009011829706754, 12.30420490275246952159703591225, 12.91022608399779005766171388392, 13.71912852598233439619246992342, 14.36688358247747045698929482047, 15.03546072888587588027377752144, 15.51580388589442689154871971671, 16.5675947690183176949531941662, 17.10079826519197458144763184672, 17.91186044744224723276544651083, 18.51345843236486150606313752509

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