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Properties

Label 1-21e2-441.209-r0-0-0
Degree $1$
Conductor $441$
Sign $0.417 + 0.908i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.733 − 0.680i)5-s + (0.222 − 0.974i)8-s + (0.222 + 0.974i)10-s + (−0.826 − 0.563i)11-s + (−0.0747 + 0.997i)13-s + (−0.733 + 0.680i)16-s + (0.623 + 0.781i)17-s − 19-s + (0.365 − 0.930i)20-s + (0.365 + 0.930i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.733 − 0.680i)5-s + (0.222 − 0.974i)8-s + (0.222 + 0.974i)10-s + (−0.826 − 0.563i)11-s + (−0.0747 + 0.997i)13-s + (−0.733 + 0.680i)16-s + (0.623 + 0.781i)17-s − 19-s + (0.365 − 0.930i)20-s + (0.365 + 0.930i)22-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.417 + 0.908i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.417 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3133658297 + 0.2008149498i\)
\(L(\frac12)\) \(\approx\) \(0.3133658297 + 0.2008149498i\)
\(L(1)\) \(\approx\) \(0.5232186496 - 0.08355758530i\)
\(L(1)\) \(\approx\) \(0.5232186496 - 0.08355758530i\)

Euler product

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

−23.73684202475145147590215803050, −23.30193637682365258918138476054, −22.54304403714837626667798892338, −21.1631296384773631936087857650, −20.174963325188034984197009484809, −19.47387219854858371578626890568, −18.564331513395757202723561336048, −17.97962518437871886126534685162, −17.06690002451620992823457956253, −15.92162459788631408040999455063, −15.36182079530573659484977357967, −14.69395190410706887529182511922, −13.56123401030989218298953137536, −12.26603848093435390485510602679, −11.25721651165553023426554439274, −10.3813020887188067410746381452, −9.741242331261041436463912234031, −8.31762280334521456148409910572, −7.71623866529059032804781150920, −6.94804603863382874507707966245, −5.79953318342548325967840516608, −4.77255717735560221562360603875, −3.24353358036590309583110932638, −2.11208871842180326976507535961, −0.299588008106431129239985413694, 1.2174838795360534295662091666, 2.49171434046179545545336822487, 3.74481399068870448572896137829, 4.60359696472833802685931399494, 6.14720486911040272542445934800, 7.37662821486259957470871184715, 8.309776443307020031213135535151, 8.79479608779952154855270238764, 10.02144184030822507585177081104, 10.89499005800796787436241070037, 11.76655645828572105999715209480, 12.57823729565010619767386072009, 13.32895530392644603315003394730, 14.78077499185363905592181977359, 15.85283012735923909070376912320, 16.61824789780652785080679066517, 17.11997532799798620229729135602, 18.58560925466634415548745843255, 18.93198050857544267019547386086, 19.84484660205145234076366525828, 20.74141360717383518465938130454, 21.31925461259595484191943234215, 22.27463779005547674493258787426, 23.69517644174931924876922532345, 24.02317160107262761052468047141

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