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Properties

Label 1-2432-2432.1595-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  + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (0.382 − 0.923i)7-s + (0.382 + 0.923i)9-s + (−0.555 − 0.831i)11-s + (−0.980 + 0.195i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.831 + 0.555i)21-s + (0.923 − 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (0.555 − 0.831i)29-s − i·31-s − i·33-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)3-s + (−0.980 − 0.195i)5-s + (0.382 − 0.923i)7-s + (0.382 + 0.923i)9-s + (−0.555 − 0.831i)11-s + (−0.980 + 0.195i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.831 + 0.555i)21-s + (0.923 − 0.382i)23-s + (0.923 + 0.382i)25-s + (0.195 − 0.980i)27-s + (0.555 − 0.831i)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: $\chi_{2432} (1595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (0:\ ),\ -0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2697980533 - 0.8166903638i\)
\(L(\frac12)\) \(\approx\) \(0.2697980533 - 0.8166903638i\)
\(L(1)\) \(\approx\) \(0.6328829664 - 0.3286644899i\)
\(L(1)\) \(\approx\) \(0.6328829664 - 0.3286644899i\)

Euler product

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

−19.83166207912440281253893240589, −18.828975331620568470476464067228, −18.53163335762049436586965257771, −17.41409514303909730460893857509, −17.15800433684747246261379302410, −16.00971126249586051542309524453, −15.59212919483878576042530534103, −14.772063586247625698510078607311, −14.67345180477902650993741969501, −12.870827820024331049338578069982, −12.492823342942954151847138388855, −11.76632616241483404263165345968, −11.26100105502005369070178734590, −10.34582573684933476467754208281, −9.795913312861206082792564544966, −8.836758511307905512522109621184, −7.96869322663876000183527405955, −7.266706746705299314286902014126, −6.44602442048931367973453618497, −5.31914271937285235039439656377, −5.007796664316571230036623088426, −4.13389044805526565213594277347, −3.20289307589950384507084551353, −2.30332960052441398622775809005, −0.98002206843620251525295344853, 0.47826864654729103864365628658, 0.969214917114365798054139969808, 2.34660656275296352157465603008, 3.33740210244117669259677364526, 4.37909949976035450929493825318, 4.97549252323804810024251479146, 5.68789147450159779418198580951, 6.98754870711031106359958883148, 7.24504426491328424858028709852, 7.99240976423964345994613756187, 8.74344831434262324926429792481, 10.04237274416347753233732818948, 10.67296237967058504778081804954, 11.32747160060131647970485887242, 11.97499066331899488329239029173, 12.56577756273955147678782519025, 13.45414427500183924912444994943, 14.071006942769953526316044306402, 14.9484941290926089439990693019, 15.9404645498422453955238046985, 16.482228123538382361040258362956, 16.993111669603318095891101260763, 17.75301444229085846028220505256, 18.60911751323417996904292650070, 19.19669141713207323884983983015

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