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Properties

Label 1-408-408.299-r1-0-0
Degree $1$
Conductor $408$
Sign $-0.855 - 0.518i$
Analytic cond. $43.8456$
Root an. cond. $43.8456$
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  + (0.923 − 0.382i)5-s + (−0.923 − 0.382i)7-s + (0.382 − 0.923i)11-s − i·13-s + (−0.707 − 0.707i)19-s + (0.382 − 0.923i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)29-s + (0.382 + 0.923i)31-s − 35-s + (−0.382 − 0.923i)37-s + (−0.923 − 0.382i)41-s + (−0.707 + 0.707i)43-s − i·47-s + (0.707 + 0.707i)49-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)5-s + (−0.923 − 0.382i)7-s + (0.382 − 0.923i)11-s − i·13-s + (−0.707 − 0.707i)19-s + (0.382 − 0.923i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)29-s + (0.382 + 0.923i)31-s − 35-s + (−0.382 − 0.923i)37-s + (−0.923 − 0.382i)41-s + (−0.707 + 0.707i)43-s − i·47-s + (0.707 + 0.707i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(408\)    =    \(2^{3} \cdot 3 \cdot 17\)
Sign: $-0.855 - 0.518i$
Analytic conductor: \(43.8456\)
Root analytic conductor: \(43.8456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{408} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 408,\ (1:\ ),\ -0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2818340470 - 1.008718427i\)
\(L(\frac12)\) \(\approx\) \(0.2818340470 - 1.008718427i\)
\(L(1)\) \(\approx\) \(0.9544833055 - 0.2604697330i\)
\(L(1)\) \(\approx\) \(0.9544833055 - 0.2604697330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−24.77010604637910995235147373631, −23.39804789266569607385842071384, −22.509302703138818234378702615673, −22.12895832842469857481046493943, −20.974380726596021135786790214595, −20.21630442356860169489072323540, −19.11217693603270811771238296621, −18.419250917095641392471076374649, −17.36533888935446813364651822964, −16.846659465503821564007335220246, −15.38293131394155188096177683695, −14.986301090996757873018590983162, −13.69943735743136843829743495683, −12.96284077547067817260309860899, −12.17072346759452033301010137528, −10.85801672966283091369424878552, −9.83880460857648670663995666841, −9.46420713194355531472151216838, −8.08073141837268245151154626459, −6.87416424669947898990053338351, −6.090960794105723219703921054301, −5.19409350892423385750464069891, −3.68921665041157670432534607170, −2.65698863482203665231348892485, −1.56612331180703928526340889517, 0.266386049888419076663516941785, 1.60463702176147961290542483539, 2.87358317986256028905459162941, 4.04980328801109490676453921785, 5.23161118640095005824370904774, 6.392376560772476823315625375312, 6.879212340331309515173215825085, 8.636732569872096532107204344005, 9.15252783010708749969350278061, 10.18401854907839210978251984389, 11.06819472639894129674196841873, 12.30618579131662537297647533251, 13.21823450742070241110809354504, 13.827602508851471050596351642226, 14.76387530368189144991481116410, 16.21585352913616434306996224178, 16.62371979372881197241868417186, 17.4562935699602259107830099971, 18.64335381616898743472001714531, 19.35569263923191864171823391222, 20.27278691656617632563948808760, 21.34290794677468512602548909843, 21.84620606209198464338825662702, 22.81874853094512540629981245039, 23.86466581257944035732238091250

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