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Properties

Label 2-21e2-49.22-c1-0-21
Degree $2$
Conductor $441$
Sign $0.142 + 0.989i$
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  + (2.37 − 1.14i)2-s + (3.09 − 3.88i)4-s + (0.398 − 1.74i)5-s + (0.566 + 2.58i)7-s + (1.74 − 7.63i)8-s + (−1.05 − 4.60i)10-s + (−1.38 + 0.664i)11-s + (−4.78 + 2.30i)13-s + (4.30 + 5.49i)14-s + (−2.38 − 10.4i)16-s + (3.34 + 4.18i)17-s − 2.34·19-s + (−5.54 − 6.95i)20-s + (−2.52 + 3.16i)22-s + (2.25 − 2.82i)23-s + ⋯
L(s)  = 1  + (1.68 − 0.809i)2-s + (1.54 − 1.94i)4-s + (0.178 − 0.780i)5-s + (0.214 + 0.976i)7-s + (0.616 − 2.69i)8-s + (−0.332 − 1.45i)10-s + (−0.416 + 0.200i)11-s + (−1.32 + 0.639i)13-s + (1.15 + 1.46i)14-s + (−0.597 − 2.61i)16-s + (0.810 + 1.01i)17-s − 0.538·19-s + (−1.24 − 1.55i)20-s + (−0.537 + 0.674i)22-s + (0.469 − 0.588i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.142 + 0.989i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.142 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63653 - 2.28314i\)
\(L(\frac12)\) \(\approx\) \(2.63653 - 2.28314i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.566 - 2.58i)T \)
good2 \( 1 + (-2.37 + 1.14i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-0.398 + 1.74i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (1.38 - 0.664i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (4.78 - 2.30i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-3.34 - 4.18i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
23 \( 1 + (-2.25 + 2.82i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (4.39 + 5.51i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 1.98T + 31T^{2} \)
37 \( 1 + (-7.41 - 9.30i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-0.667 + 2.92i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.38 + 10.4i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (7.07 - 3.40i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.538 + 0.675i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.13 + 4.98i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-4.29 - 5.39i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 8.90T + 67T^{2} \)
71 \( 1 + (5.10 - 6.39i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-5.23 - 2.52i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 3.91T + 79T^{2} \)
83 \( 1 + (-2.93 - 1.41i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (7.35 + 3.53i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 3.04T + 97T^{2} \)
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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

−11.34900153732846381986605695413, −10.23861406234256649907368958871, −9.451850459183215207191463670819, −8.197433282862574215627837992637, −6.74165542452049460055196461330, −5.64944359088998579813939142046, −5.03627059523967276268613509225, −4.17503449283967186019848225541, −2.71413149089488798633978645885, −1.78575542572458640272794275336, 2.65247842070087857172872564710, 3.48683346617475603013461697470, 4.75886902879150673182380418185, 5.44547297237839465263194196941, 6.63557961382707859802330372932, 7.39598122971644507331620990818, 7.84073854596978240890396851326, 9.687980384892357386678937729349, 10.79746480516229879846794044967, 11.44150328115945244004386199715

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