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Properties

Label 2-21e2-63.59-c1-0-18
Degree $2$
Conductor $441$
Sign $0.935 - 0.354i$
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  + (1.02 + 0.589i)2-s + (−1.34 + 1.09i)3-s + (−0.305 − 0.529i)4-s + 4.33·5-s + (−2.01 + 0.327i)6-s − 3.07i·8-s + (0.599 − 2.93i)9-s + (4.42 + 2.55i)10-s − 2.16i·11-s + (0.990 + 0.375i)12-s + (2.25 + 1.30i)13-s + (−5.81 + 4.74i)15-s + (1.20 − 2.08i)16-s + (0.585 − 1.01i)17-s + (2.34 − 2.64i)18-s + (−2.09 + 1.20i)19-s + ⋯
L(s)  = 1  + (0.721 + 0.416i)2-s + (−0.774 + 0.632i)3-s + (−0.152 − 0.264i)4-s + 1.93·5-s + (−0.822 + 0.133i)6-s − 1.08i·8-s + (0.199 − 0.979i)9-s + (1.39 + 0.807i)10-s − 0.651i·11-s + (0.286 + 0.108i)12-s + (0.624 + 0.360i)13-s + (−1.50 + 1.22i)15-s + (0.300 − 0.520i)16-s + (0.142 − 0.245i)17-s + (0.552 − 0.623i)18-s + (−0.480 + 0.277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.935 - 0.354i$
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: \((2,\ 441,\ (\ :1/2),\ 0.935 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98728 + 0.364117i\)
\(L(\frac12)\) \(\approx\) \(1.98728 + 0.364117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 - 1.09i)T \)
7 \( 1 \)
good2 \( 1 + (-1.02 - 0.589i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 4.33T + 5T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + (-2.25 - 1.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.585 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.09 - 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.65iT - 23T^{2} \)
29 \( 1 + (-0.589 + 0.340i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.67 - 3.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.55 - 4.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.68 - 6.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.57 + 6.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.79 + 1.61i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.91 - 5.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.21 + 3.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.32 + 5.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.95iT - 71T^{2} \)
73 \( 1 + (10.3 + 5.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.87 + 8.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.796 - 1.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.04 - 5.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.36 - 1.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.92825222682770170250355082738, −10.26153667084946614117788144817, −9.531603750528886065418444363858, −8.863777845389338690898447285853, −6.78936433431212109119284004917, −6.12055289586294352579918137634, −5.55668667578434408137396891022, −4.78738683473243146238463074329, −3.42598329225086674094309286789, −1.42007938824275365801261532084, 1.71106212678557602056881908314, 2.63317994749251860029902313076, 4.43512380369749136746803866201, 5.46271697550161389422672208760, 6.02223868159332144768348065370, 7.07181974783991572472693289760, 8.381814594833049355100256171896, 9.402485867397941242778573356872, 10.49641729164250875499361537760, 11.03947298213197755632168317507

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