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Properties

Label 1-21e2-441.250-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.700 + 0.713i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)5-s + (0.623 − 0.781i)8-s + (0.988 − 0.149i)10-s + (−0.222 + 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 − 0.997i)26-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)5-s + (0.623 − 0.781i)8-s + (0.988 − 0.149i)10-s + (−0.222 + 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 − 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.700 + 0.713i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3051237070 + 0.7274558042i\)
\(L(\frac12)\) \(\approx\) \(0.3051237070 + 0.7274558042i\)
\(L(1)\) \(\approx\) \(0.6543133797 + 0.1179717093i\)
\(L(1)\) \(\approx\) \(0.6543133797 + 0.1179717093i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 + 0.997i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.733 - 0.680i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−23.5977262648966797316023019681, −23.12402936861628120689407846000, −21.85825059422032174150568677160, −20.594286595814863321903013385004, −20.01132795031067071929656725919, −19.14336969150122208836010009372, −18.25030361188786611843229293715, −17.47880903595686157229926475701, −16.335698329584137152158837871880, −15.94694098023555854022788291291, −15.20046814917594973415132635930, −13.842194094658458230151839189911, −13.25033372307746115444016866286, −11.72263187927181432667581707538, −11.12987591745406581000331426793, −9.89115000757035855569033757603, −8.97864319702627800809308611034, −8.165590077237128172747921739500, −7.526343370658938951165710517288, −6.18726622735493761581898894655, −5.363235281677519621012252942747, −4.30002601092947164428821258433, −2.8141281814634947720904436810, −1.04949530379562228433441068453, −0.33506593005732368557680630267, 1.37559051814287928935315970676, 2.47079266349495895924668991542, 3.64258784392590574178011085805, 4.37919774545750623158790213377, 6.25024572203301056152705726458, 7.196217885427423947866729163287, 8.051299251376585611091413281986, 8.93376018721833156003666337235, 10.31019559464951899462683516204, 10.52478388122340933316851021377, 11.89219476276281557840529419266, 12.23345137547807300881602298424, 13.55399935891504932199605585914, 14.58447373296424984879211688055, 15.67267705742214099499614082999, 16.37999901283632130674884882653, 17.5654799022521700401954438701, 18.27388380113616987141452940911, 18.9585957571082210306188895599, 19.8049044407952822386303690037, 20.56091610187535065925969270961, 21.49706680975631743010955477900, 22.361726466698015107283322866831, 23.14573446588051769523154129428, 24.06747999027524340502338734442

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