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Properties

Label 2-21e2-9.4-c1-0-27
Degree $2$
Conductor $441$
Sign $-0.190 + 0.981i$
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  + (−0.119 − 0.207i)2-s + (−1.12 + 1.31i)3-s + (0.971 − 1.68i)4-s + (1.29 − 2.24i)5-s + (0.407 + 0.0753i)6-s − 0.942·8-s + (−0.471 − 2.96i)9-s − 0.619·10-s + (−2.09 − 3.62i)11-s + (1.12 + 3.17i)12-s + (−1.84 + 3.18i)13-s + (1.5 + 4.23i)15-s + (−1.83 − 3.16i)16-s + 1.71·17-s + (−0.557 + 0.451i)18-s − 7.15·19-s + ⋯
L(s)  = 1  + (−0.0845 − 0.146i)2-s + (−0.649 + 0.760i)3-s + (0.485 − 0.841i)4-s + (0.579 − 1.00i)5-s + (0.166 + 0.0307i)6-s − 0.333·8-s + (−0.157 − 0.987i)9-s − 0.195·10-s + (−0.630 − 1.09i)11-s + (0.324 + 0.915i)12-s + (−0.510 + 0.884i)13-s + (0.387 + 1.09i)15-s + (−0.457 − 0.792i)16-s + 0.414·17-s + (−0.131 + 0.106i)18-s − 1.64·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.667594 - 0.809271i\)
\(L(\frac12)\) \(\approx\) \(0.667594 - 0.809271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.12 - 1.31i)T \)
7 \( 1 \)
good2 \( 1 + (0.119 + 0.207i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.29 + 2.24i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.09 + 3.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.84 - 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.71T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
23 \( 1 + (-2.56 + 4.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.06 - 1.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.26 + 5.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.66T + 37T^{2} \)
41 \( 1 + (-5.10 + 8.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.66 + 8.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + (3.03 - 5.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.99 - 6.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.13 - 7.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.23T + 71T^{2} \)
73 \( 1 - 7.15T + 73T^{2} \)
79 \( 1 + (-4.91 - 8.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.44 + 5.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.03T + 89T^{2} \)
97 \( 1 + (1.53 + 2.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72339049645176900711895601936, −10.14878771870965005446638620562, −9.182611857039085434902873297730, −8.568378216973057835061421282383, −6.78734447921161680834309417609, −5.89406321738283592058025149106, −5.23537841973169423553731731007, −4.25745386272669393301651598459, −2.40508192808339687135047845696, −0.70235747432514382084825069620, 2.12691006061429817489875034742, 2.95349928948755253083014340954, 4.78352172001413331890786506017, 6.06534244789415975468425256851, 6.77932298876229540801415761465, 7.53870733953334530733643396424, 8.224454631597656717979996416737, 9.842315860753791562534312189817, 10.63618111383023209141904740490, 11.29741658813850605579628241733

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