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Properties

Label 1-2432-2432.531-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: $\chi_{2432} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (0:\ ),\ -0.803 + 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1576755259 + 0.4772906292i\)
\(L(\frac12)\) \(\approx\) \(0.1576755259 + 0.4772906292i\)
\(L(1)\) \(\approx\) \(0.9031566978 + 0.1033028194i\)
\(L(1)\) \(\approx\) \(0.9031566978 + 0.1033028194i\)

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

−19.14225381321466256350324517712, −18.49726821109885892933255800936, −17.9799956212709656830735972624, −17.56809586999493766026503221202, −16.05722660596492298672339800560, −15.55512876118928687269486243480, −14.920802864583432529715214540440, −14.19543787766966812876353386588, −13.57525497473179059686039680963, −12.53009036072906603575986320156, −12.27102813806389176377052684422, −11.44198417122442871613405335075, −10.24633833986517213902482739402, −9.940820816598119256421661303384, −8.755462109934793705009171419488, −8.161240768684536626671081748207, −7.26145121381433730818307644924, −6.94792163977161999907236833068, −5.77608850421096083581946803904, −5.3455292969917811531875544779, −3.823964084055614281181935758222, −2.880227333617149075760166766920, −2.67870656287392214390543382295, −1.6559727249757175204025387796, −0.14853248318493429889447512107, 1.17579026678114818036912293540, 2.300929631388558450541191019452, 3.29181572597538844921017420070, 4.17372843791118661788257098842, 4.4849265330688203777778418795, 5.52527000604970013036105938597, 6.31671794127766153805871240357, 7.685685926140088542398245125, 7.9997007275869105137053007872, 8.87140563135885069236265818997, 9.61535847419660767554011919910, 10.231157843101657142935937199483, 10.90885472164135833624921841306, 11.86817381941873854822521834704, 12.66907363229748916465073672884, 13.61833149334130231592129436633, 13.84721794630090080883834245554, 14.826864318802279295772210625356, 15.70515900912646110839567964730, 16.28821718789562707292725800668, 16.7077747051695796896680090991, 17.32853679676226522320385491275, 18.5725557195943916529195434468, 19.4213872384490219079927651087, 19.74362938613526906829412670655

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