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Properties

Label 2-21e2-63.4-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.321 - 0.946i$
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  + (−0.849 + 1.47i)2-s + (1.64 + 0.545i)3-s + (−0.444 − 0.769i)4-s + 3.58·5-s + (−2.19 + 1.95i)6-s − 1.88·8-s + (2.40 + 1.79i)9-s + (−3.04 + 5.28i)10-s − 2.81·11-s + (−0.310 − 1.50i)12-s + (0.5 − 0.866i)13-s + (5.89 + 1.95i)15-s + (2.49 − 4.31i)16-s + (−2.05 + 3.56i)17-s + (−4.68 + 2.01i)18-s + (−0.444 − 0.769i)19-s + ⋯
L(s)  = 1  + (−0.600 + 1.04i)2-s + (0.949 + 0.314i)3-s + (−0.222 − 0.384i)4-s + 1.60·5-s + (−0.897 + 0.798i)6-s − 0.667·8-s + (0.801 + 0.597i)9-s + (−0.964 + 1.67i)10-s − 0.847·11-s + (−0.0897 − 0.435i)12-s + (0.138 − 0.240i)13-s + (1.52 + 0.505i)15-s + (0.623 − 1.07i)16-s + (−0.498 + 0.863i)17-s + (−1.10 + 0.475i)18-s + (−0.101 − 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.321 - 0.946i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00732 + 1.40648i\)
\(L(\frac12)\) \(\approx\) \(1.00732 + 1.40648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.545i)T \)
7 \( 1 \)
good2 \( 1 + (0.849 - 1.47i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.58T + 5T^{2} \)
11 \( 1 + 2.81T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.05 - 3.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.444 + 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.87T + 23T^{2} \)
29 \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.38 + 4.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.70 - 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.33 - 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0618 + 0.107i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.43 + 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.93 + 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.15 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + (5.32 - 9.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.54 + 6.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.05 + 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.80 - 8.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.91091280475049956130379159059, −10.11918296850239882064600201788, −9.310483852626057564255977340087, −8.735991064337255141622895876058, −7.84483424097878508147925651294, −6.86656705964619515036794135670, −5.92076970703499884243732237081, −4.97829203422949946535368618935, −3.16426476275751591026397424429, −2.03758374673598009905404497514, 1.39721450364255502665417731466, 2.37389427422714551351038888063, 3.13067244034551400339363084194, 5.02859251282328740709304202641, 6.24071304491572843407352867369, 7.23492811769765037725048789410, 8.752126724188490946768148497311, 9.055458304498382951474858406506, 10.05374280456957810548583490556, 10.44280679241410389134037925909

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