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Properties

Label 1-4641-4641.164-r0-0-0
Degree $1$
Conductor $4641$
Sign $0.0275 + 0.999i$
Analytic cond. $21.5527$
Root an. cond. $21.5527$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (0.130 − 0.991i)5-s + (−0.707 − 0.707i)8-s + (0.991 − 0.130i)10-s + (0.991 − 0.130i)11-s + (0.5 − 0.866i)16-s + (0.258 + 0.965i)19-s + (0.382 + 0.923i)20-s + (0.382 + 0.923i)22-s + (0.793 − 0.608i)23-s + (−0.965 − 0.258i)25-s + (0.382 + 0.923i)29-s + (−0.608 + 0.793i)31-s + (0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (0.130 − 0.991i)5-s + (−0.707 − 0.707i)8-s + (0.991 − 0.130i)10-s + (0.991 − 0.130i)11-s + (0.5 − 0.866i)16-s + (0.258 + 0.965i)19-s + (0.382 + 0.923i)20-s + (0.382 + 0.923i)22-s + (0.793 − 0.608i)23-s + (−0.965 − 0.258i)25-s + (0.382 + 0.923i)29-s + (−0.608 + 0.793i)31-s + (0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0275 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0275 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4641\)    =    \(3 \cdot 7 \cdot 13 \cdot 17\)
Sign: $0.0275 + 0.999i$
Analytic conductor: \(21.5527\)
Root analytic conductor: \(21.5527\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4641} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4641,\ (0:\ ),\ 0.0275 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317354654 + 1.281560242i\)
\(L(\frac12)\) \(\approx\) \(1.317354654 + 1.281560242i\)
\(L(1)\) \(\approx\) \(1.077580753 + 0.4820938015i\)
\(L(1)\) \(\approx\) \(1.077580753 + 0.4820938015i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.130 - 0.991i)T \)
11 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.793 - 0.608i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (-0.608 + 0.793i)T \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.258 - 0.965i)T \)
61 \( 1 + (0.608 + 0.793i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.793 + 0.608i)T \)
79 \( 1 + (-0.793 + 0.608i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−18.12284185905913506516205370136, −17.48090968377383618514933226945, −17.0722379395355810060881104508, −15.79432977688005351083751623150, −15.07086976444114584071516262498, −14.64894344561158345172917261989, −13.73864407369735635914769443639, −13.53272912877684479176853416015, −12.50725193514139734089409549797, −11.80974678644545431453416290042, −11.2269908959586063937397625267, −10.77234060213609853179353158320, −9.86235295248112364830253572118, −9.38938619813919291401974449070, −8.71784015959164815215042129414, −7.64047117059543661749618602995, −6.88937709131486643891673263044, −6.165577377345057003989721067281, −5.39657936357036355912098390838, −4.5172262617695581974551773266, −3.73879174351568020224719565107, −3.15494092969569020811445725073, −2.33998730101609841834533856389, −1.65282663356126877344960219534, −0.56913385179368852884900914609, 0.89117907345176447449190426961, 1.59603862801442411843382383560, 2.99386597488897264294752198913, 3.827418966502403403285785179265, 4.43053428384124338008149453259, 5.25399036261178403997550220089, 5.72407276572578104007495318087, 6.65671997800903171318275444820, 7.15713981853995394970850112774, 8.20816711342324804576478225777, 8.621187676541132571051954861999, 9.26219445479830401435496065179, 9.89201899215577183421493916690, 10.89970068390353102449916823185, 11.983844218289815721086773974574, 12.39020693725451508617128611745, 13.00623637563466133453239390653, 13.80631952745606849233048632616, 14.393532318635651849785168047359, 14.902761908095355992445518769151, 15.93623703306376972416252113, 16.31153711292757495205368184099, 16.90109169837739958660313064579, 17.43953224077895765227543714038, 18.15642228589826187903647136256

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