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Properties

Label 2-21e2-63.20-c1-0-21
Degree $2$
Conductor $441$
Sign $0.391 - 0.920i$
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  + (1.58 + 0.916i)2-s + (1.44 + 0.958i)3-s + (0.678 + 1.17i)4-s + (−0.322 − 0.559i)5-s + (1.41 + 2.84i)6-s − 1.17i·8-s + (1.16 + 2.76i)9-s − 1.18i·10-s + (4.60 + 2.65i)11-s + (−0.147 + 2.34i)12-s + (−4.44 + 2.56i)13-s + (0.0702 − 1.11i)15-s + (2.43 − 4.22i)16-s − 1.62·17-s + (−0.689 + 5.45i)18-s − 2.41i·19-s + ⋯
L(s)  = 1  + (1.12 + 0.647i)2-s + (0.832 + 0.553i)3-s + (0.339 + 0.587i)4-s + (−0.144 − 0.250i)5-s + (0.575 + 1.16i)6-s − 0.416i·8-s + (0.387 + 0.921i)9-s − 0.374i·10-s + (1.38 + 0.801i)11-s + (−0.0425 + 0.676i)12-s + (−1.23 + 0.711i)13-s + (0.0181 − 0.288i)15-s + (0.609 − 1.05i)16-s − 0.395·17-s + (−0.162 + 1.28i)18-s − 0.553i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54065 + 1.67988i\)
\(L(\frac12)\) \(\approx\) \(2.54065 + 1.67988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.44 - 0.958i)T \)
7 \( 1 \)
good2 \( 1 + (-1.58 - 0.916i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.322 + 0.559i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.60 - 2.65i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.44 - 2.56i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (1.27 - 0.735i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.43 + 3.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.90 + 2.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.99T + 37T^{2} \)
41 \( 1 + (5.99 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.51 - 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.54 + 2.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.36iT - 53T^{2} \)
59 \( 1 + (1.47 + 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.18 - 5.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + (3.48 - 6.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.51 - 6.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 + (-14.3 - 8.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.63339133948719489394371431456, −10.07015342637174436951013172345, −9.503862407355034878189134292528, −8.598737447292998019227609655871, −7.24600380738495662477794418565, −6.75000510706529174913818547961, −5.24999913791425435986243389392, −4.39680919773070546260349932262, −3.83748110969234911352382850290, −2.22020047206875066740338809471, 1.70628187003360257683182945526, 3.06908210108307382863255062254, 3.64728961562049205378131203473, 4.91085663261677639588434765809, 6.17480377332962452641688528018, 7.17280236744875217097719561296, 8.249918914071582326080411243885, 9.074476465365693760829800002781, 10.20855118765794350448524588726, 11.34708942868424051127889576741

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