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Properties

Label 2-21e2-63.59-c1-0-21
Degree $2$
Conductor $441$
Sign $0.578 + 0.815i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 + 0.742i)2-s + (−1.72 − 0.153i)3-s + (0.101 + 0.176i)4-s − 0.308·5-s + (−2.10 − 1.47i)6-s − 2.66i·8-s + (2.95 + 0.528i)9-s + (−0.396 − 0.228i)10-s − 3.16i·11-s + (−0.148 − 0.320i)12-s + (−3.00 − 1.73i)13-s + (0.532 + 0.0472i)15-s + (2.18 − 3.78i)16-s + (2.44 − 4.22i)17-s + (3.40 + 2.87i)18-s + (4.62 − 2.67i)19-s + ⋯
L(s)  = 1  + (0.909 + 0.524i)2-s + (−0.996 − 0.0884i)3-s + (0.0509 + 0.0882i)4-s − 0.137·5-s + (−0.859 − 0.603i)6-s − 0.942i·8-s + (0.984 + 0.176i)9-s + (−0.125 − 0.0723i)10-s − 0.953i·11-s + (−0.0429 − 0.0923i)12-s + (−0.833 − 0.481i)13-s + (0.137 + 0.0121i)15-s + (0.545 − 0.945i)16-s + (0.592 − 1.02i)17-s + (0.802 + 0.676i)18-s + (1.06 − 0.612i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.578 + 0.815i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.578 + 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19654 - 0.618350i\)
\(L(\frac12)\) \(\approx\) \(1.19654 - 0.618350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.153i)T \)
7 \( 1 \)
good2 \( 1 + (-1.28 - 0.742i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.308T + 5T^{2} \)
11 \( 1 + 3.16iT - 11T^{2} \)
13 \( 1 + (3.00 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 + 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.97iT - 23T^{2} \)
29 \( 1 + (-2.70 + 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.51 + 3.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.92 + 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.58 - 4.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.23 - 7.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0740 + 0.0427i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.04 - 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.69 - 2.71i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0554 - 0.0959i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.78iT - 71T^{2} \)
73 \( 1 + (8.32 + 4.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.56 - 4.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.42 - 7.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.936 - 1.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.34068737735270522029965009084, −10.02538138500752299926585762782, −9.513423617049176780054592327678, −7.74369196149340899385958912567, −7.09908406497610169052075668171, −5.90556770439386456339945141591, −5.40741177784142541875987661594, −4.49850205094562210080815104467, −3.19554052105018132782885873534, −0.75254406603239069910564064303, 1.87014420398682851602314559823, 3.50337038399964181387930884321, 4.56836771133803467927661227748, 5.17530459171682477240918801840, 6.34530173468848588727454783369, 7.39255915537370143577104751817, 8.477509784147494106103935561324, 10.01283021312854380741630419850, 10.36807362577298247731488777912, 11.79334040592450531732912717363

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