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Properties

Label 1-448-448.101-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.955 + 0.294i$
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  + (0.608 − 0.793i)3-s + (−0.793 + 0.608i)5-s + (−0.258 − 0.965i)9-s + (−0.991 + 0.130i)11-s + (0.923 − 0.382i)13-s + i·15-s + (0.866 − 0.5i)17-s + (0.130 − 0.991i)19-s + (0.258 + 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.923 − 0.382i)27-s + (0.382 + 0.923i)29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.793 + 0.608i)37-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)3-s + (−0.793 + 0.608i)5-s + (−0.258 − 0.965i)9-s + (−0.991 + 0.130i)11-s + (0.923 − 0.382i)13-s + i·15-s + (0.866 − 0.5i)17-s + (0.130 − 0.991i)19-s + (0.258 + 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.923 − 0.382i)27-s + (0.382 + 0.923i)29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.793 + 0.608i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.955 + 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05550970081 - 0.3689407831i\)
\(L(\frac12)\) \(\approx\) \(-0.05550970081 - 0.3689407831i\)
\(L(1)\) \(\approx\) \(0.8895647157 - 0.2409906112i\)
\(L(1)\) \(\approx\) \(0.8895647157 - 0.2409906112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.608 - 0.793i)T \)
5 \( 1 + (-0.793 + 0.608i)T \)
11 \( 1 + (-0.991 + 0.130i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.130 - 0.991i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.793 + 0.608i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.991 + 0.130i)T \)
59 \( 1 + (-0.130 - 0.991i)T \)
61 \( 1 + (-0.991 - 0.130i)T \)
67 \( 1 + (-0.608 + 0.793i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.965 - 0.258i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (0.965 - 0.258i)T \)
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

−24.24563850253877180855659574347, −23.33134907699755213646677598102, −22.69555765883932748063407813790, −21.26810162952520828559651306354, −20.94142661320427949470791928338, −20.17783813785894052535757751795, −19.124288797681558246652787115145, −18.58216833221244129432238321258, −17.04040951983722662778029056886, −16.19293143926940080857128331241, −15.772154647080950614926602388207, −14.782990737848933162459154576616, −13.88920276096920045541272590665, −12.88813794101112799608368262690, −11.95293901564504722348889690284, −10.78203049880714527966960637792, −10.15214939256587009685910446417, −8.83691981535194550486461107419, −8.31882947363972465259445815593, −7.47260874372959525277329074642, −5.82824153968301976699010254302, −4.85831174730109996648278028384, −3.887145872969541022817114133739, −3.11966934671431905001849969167, −1.59431096980984718142311351891, 0.08999905859207553216574115552, 1.43610000062465562158171005714, 2.96665375341500324763864592027, 3.3450962670397629741230629247, 4.938640394563178200953655259506, 6.23186690566686091259078406412, 7.27078919408517300290836250701, 7.84256600341177834053287787516, 8.73097253497320050729068280212, 9.94819565723532180812501925619, 11.09084804447802483229671911643, 11.83142717778900542939811646266, 12.9504358614471800855955876506, 13.57368330201616099989507496281, 14.636770175596803043476154317, 15.39525899992975950361451946518, 16.140073906336520904278088216121, 17.63096934462669825240622773556, 18.38833815278955724374055600466, 18.88997399840896482721930214086, 19.88669488740148246427275332206, 20.522387123570902666828106114674, 21.5257814790876111214011047313, 22.77727318835814174451938066402, 23.508323431670134043208621424454

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