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Properties

Label 1-2432-2432.1747-r0-0-0
Degree $1$
Conductor $2432$
Sign $0.803 - 0.595i$
Analytic cond. $11.2941$
Root an. cond. $11.2941$
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  + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 + 0.831i)21-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 + 0.831i)21-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(11.2941\)
Root analytic conductor: \(11.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (0:\ ),\ 0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248000552 - 0.4122836934i\)
\(L(\frac12)\) \(\approx\) \(1.248000552 - 0.4122836934i\)
\(L(1)\) \(\approx\) \(0.9223776759 - 0.1648705890i\)
\(L(1)\) \(\approx\) \(0.9223776759 - 0.1648705890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.555 - 0.831i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.831 + 0.555i)T \)
13 \( 1 + (0.195 - 0.980i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.555 - 0.831i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.831 - 0.555i)T \)
59 \( 1 + (-0.195 - 0.980i)T \)
61 \( 1 + (0.555 + 0.831i)T \)
67 \( 1 + (0.555 + 0.831i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.980 + 0.195i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
97 \( 1 - iT \)
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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

−19.67997436550156442136196320128, −18.84481873195707244016337542314, −18.10700289514035499470652688738, −17.20442957385689396567590577198, −16.646007998894530208622929399437, −16.04574246780576126518878464026, −15.67249269615142325024402647061, −14.51613506389797275887733638591, −14.01887968715151755322012380290, −12.96866224919100142966562292035, −12.07289704853770203085280587614, −11.79843876545134042199586188004, −11.03628606247768070916902741821, −9.807315455564616557680550533263, −9.37364262218377227232019083478, −8.93254979202230202694804145918, −8.03920550925816226114157632083, −6.75487064623685370226185240597, −5.842897387247823142248092565483, −5.6172006170637096804249268374, −4.53185021316094475207622159925, −3.95116169509561297330262602035, −2.99838734934929010085668430812, −1.80353520961063356929150750555, −0.76338114467335215474058018847, 0.676880745439637102383219903659, 1.651540809771103848864227598728, 2.55083018651162586518132154317, 3.56540255681667286207889814901, 4.25964026827585231739092534030, 5.62467491517258222705300022663, 6.14443281696394143372058472493, 6.81406507644464081456707370288, 7.61151846376545133633456163380, 7.96911178215999016097179964085, 9.422878694065687539470262316715, 10.14948533442978392599023224526, 10.78810943931568246430410084429, 11.359131837457763335991032920184, 12.3769404914258394403704950384, 12.87364227085724671469877096447, 13.67684604041073120371880184018, 14.39108775250247910308127829659, 14.91500579816583247994911985882, 16.03915332420627981263818106048, 16.80222637700285109930378133842, 17.45199423482058209331000748583, 17.900221475915580229993811244574, 18.743498061146835154142538676615, 19.30955169586194152883032534946

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