Properties

Label 1-2432-2432.1291-r0-0-0
Degree $1$
Conductor $2432$
Sign $0.514 + 0.857i$
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  + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-s + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.195 + 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (0.980 + 0.195i)21-s + (−0.382 − 0.923i)23-s + (−0.382 + 0.923i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s i·31-s i·33-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-s + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.195 + 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (0.980 + 0.195i)21-s + (−0.382 − 0.923i)23-s + (−0.382 + 0.923i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s i·31-s i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(11.2941\)
Root analytic conductor: \(11.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (0:\ ),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.720171632 + 0.9744651931i\)
\(L(\frac12)\) \(\approx\) \(1.720171632 + 0.9744651931i\)
\(L(1)\) \(\approx\) \(1.362537383 + 0.09260421673i\)
\(L(1)\) \(\approx\) \(1.362537383 + 0.09260421673i\)

Euler product

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

−19.50749274480263028564324935307, −18.72244158236013561044101414135, −18.15549542116375941970038872754, −17.38503463122109257310852684901, −16.38303721867174326865563527119, −15.53681217924398252641938287812, −15.10766945618294210644200204367, −14.46846612291726864729106989590, −13.58783313746996859805553310242, −13.40378340684059761401989607679, −11.9840564995764741891770410264, −11.3562967643412760714996365615, −10.657827051055433819296657509439, −9.983755984921616325554672184125, −9.10704630973650655038463297786, −8.012922776993592381221217411943, −7.89985470999841229751780351080, −7.113877300070051697543850516328, −6.06319681975738496972784870407, −4.97488508593750769283984708928, −4.20921000530118876703603919851, −3.40658114216294232753206931413, −2.71726679650961385542016182701, −1.92882777413069047389627427715, −0.53830501121527924791510775519, 1.324298090359214342465967181572, 1.91314672610273437211409819865, 2.726132575307160512319696300273, 4.002850173589153325825244800830, 4.52899869950760420962997506970, 5.096007016969660534538178765217, 6.50624053926653836800940073437, 7.26674172116628660191243475221, 8.025337969732721293619588195904, 8.58293208764394355578922125142, 9.165567848036633479576255554462, 9.98644744525467951169642479491, 10.97510756715142965751860779811, 11.92235577915657649334695695263, 12.48443275387486891714340520950, 13.01943609871313022172524120589, 14.01892037917800867667660423295, 14.81834003023956953329448230980, 15.02643548902658328347348406753, 16.03670645876865183915429022889, 16.635308260641710403217897559680, 17.76999610845265948943426304785, 18.10576682121359478835790067303, 19.0889022582302002407656561717, 19.921817381300691969936205411211

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