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Properties

Label 1-448-448.237-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.0980 - 0.995i$
Analytic cond. $48.1442$
Root an. cond. $48.1442$
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  + (0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s + (0.923 + 0.382i)13-s − i·15-s − i·17-s + (−0.923 − 0.382i)19-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + 31-s + 33-s + (0.923 − 0.382i)37-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.707 − 0.707i)9-s + (0.382 + 0.923i)11-s + (0.923 + 0.382i)13-s − i·15-s − i·17-s + (−0.923 − 0.382i)19-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + 31-s + 33-s + (0.923 − 0.382i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.0980 - 0.995i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.0980 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.938656607 - 2.138977947i\)
\(L(\frac12)\) \(\approx\) \(1.938656607 - 2.138977947i\)
\(L(1)\) \(\approx\) \(1.378110354 - 0.6355380155i\)
\(L(1)\) \(\approx\) \(1.378110354 - 0.6355380155i\)

Euler product

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

−24.077940162915754331038894239844, −22.91090141293141595151981253851, −22.16898232066773654234694840392, −21.26376764658640781317555185294, −21.01693868144767882096477351150, −19.7665677749814089848321250147, −18.95863349158012279004488688241, −17.95224191289123940340507834871, −16.89325096517905599956157941689, −16.37733214696531652661483370795, −15.12027555124962667144138849183, −14.5619216632894781446945310680, −13.64123044406961001689800417972, −12.882020981438080408497808339561, −11.26410259478769824921287958863, −10.62825485448730971987237115868, −9.90603757644623722179210374378, −8.73528383617978901911662599518, −8.2888273758622831216105148340, −6.46496734287808708348992971108, −5.89439059998651718256975935233, −4.67352739496626171110207698834, −3.52531039427858702079169461133, −2.70241771561430296708326456732, −1.29688373716385013437675028550, 0.7690219435078756029485223503, 1.81102442726263277275092924622, 2.66038498724629479111012935686, 4.15597481627119645353117633173, 5.40369543497053133378579851775, 6.474259559070554738416874556, 7.10668725535681563451465122352, 8.42758594342924183841764493938, 9.12872058289031490776119868116, 10.00977050518171082525687900346, 11.40116636498926441333075119741, 12.23869928959521038480248198588, 13.281854589257375710902679608663, 13.65566645720939683343587949441, 14.66328368778549869337980340039, 15.662033615904812572817873562866, 17.02269507068185610636780106991, 17.51867325192777021526966937722, 18.39532240560682276940295986588, 19.18861976698726173069055768864, 20.25738974901122512867329429516, 20.81020808295323136425092769728, 21.746219823270014629833913858395, 23.0097959781826092436061322798, 23.50781435833675781595259973269

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