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Properties

Label 1-448-448.443-r1-0-0
Degree $1$
Conductor $448$
Sign $0.935 - 0.354i$
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  + (0.991 + 0.130i)3-s + (−0.130 − 0.991i)5-s + (0.965 + 0.258i)9-s + (0.608 + 0.793i)11-s + (0.923 − 0.382i)13-s − i·15-s + (0.866 + 0.5i)17-s + (−0.793 − 0.608i)19-s + (0.965 + 0.258i)23-s + (−0.965 + 0.258i)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 + (0.130 + 0.991i)37-s + ⋯
L(s)  = 1  + (0.991 + 0.130i)3-s + (−0.130 − 0.991i)5-s + (0.965 + 0.258i)9-s + (0.608 + 0.793i)11-s + (0.923 − 0.382i)13-s − i·15-s + (0.866 + 0.5i)17-s + (−0.793 − 0.608i)19-s + (0.965 + 0.258i)23-s + (−0.965 + 0.258i)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 + (0.130 + 0.991i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.935 - 0.354i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ 0.935 - 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.337538375 - 0.6108631396i\)
\(L(\frac12)\) \(\approx\) \(3.337538375 - 0.6108631396i\)
\(L(1)\) \(\approx\) \(1.698466313 - 0.1614655844i\)
\(L(1)\) \(\approx\) \(1.698466313 - 0.1614655844i\)

Euler product

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

−23.80543072844659081206017188972, −23.084011245334319986181267480180, −22.00783393894118025186200581955, −21.22763516029022484754944254580, −20.46706864905794537324113058612, −19.30326558145927415029881887881, −18.831844515698597920827098116506, −18.22643927231220727601499076262, −16.7990478294082717244127401609, −15.93725992155015402746657410961, −14.83158985258994317364354840486, −14.35707592960489839270715673808, −13.57691601735636690477171503248, −12.54058369747664717504582245338, −11.32395407332523395216798186112, −10.581814677114980126827287182, −9.41613110083320439434795432661, −8.63289430280723027950439291928, −7.61939347735244688077309946773, −6.77075939627869593142876080995, −5.80963033097888207021578578960, −4.02012195061752569636833810468, −3.41969946168640093654227482833, −2.37793171019495595665718229505, −1.09220690101753291397213359011, 1.0026960115701822027813320121, 1.96424090556558547088039419660, 3.41855381825601754756437575193, 4.23491276729254010767516585727, 5.236815244539506974037063570063, 6.62994458419084165850772577660, 7.77996837674679072115453288477, 8.56330915455747546694410283149, 9.28415846348025248638044880115, 10.17164418866105924615891645881, 11.415882577607481944919921295969, 12.69660647313504100666217303327, 13.04697082168894270476628804159, 14.17691432749836209898149641134, 15.105010495302691867047820867142, 15.75241777899397876467781318229, 16.8074928052937248845466930264, 17.60099657903070154856322858043, 18.88820577960435871872286496553, 19.55359626098880456966304356814, 20.416130531770354687062819523238, 20.94029572770178397946691160565, 21.769652764049735637684638254729, 23.09251108862247803710279528103, 23.77682962109054384029069263860

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