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Properties

Label 2-21e2-63.25-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.0901 - 0.995i$
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  + 1.29·2-s + (1.47 + 0.910i)3-s − 0.310·4-s + (−1.76 + 3.05i)5-s + (1.91 + 1.18i)6-s − 3.00·8-s + (1.34 + 2.68i)9-s + (−2.29 + 3.96i)10-s + (−0.589 − 1.02i)11-s + (−0.457 − 0.282i)12-s + (1.61 + 2.78i)13-s + (−5.37 + 2.89i)15-s − 3.28·16-s + (2.45 − 4.24i)17-s + (1.74 + 3.48i)18-s + (3.43 + 5.94i)19-s + ⋯
L(s)  = 1  + 0.919·2-s + (0.850 + 0.525i)3-s − 0.155·4-s + (−0.788 + 1.36i)5-s + (0.782 + 0.482i)6-s − 1.06·8-s + (0.447 + 0.894i)9-s + (−0.724 + 1.25i)10-s + (−0.177 − 0.307i)11-s + (−0.132 − 0.0815i)12-s + (0.446 + 0.773i)13-s + (−1.38 + 0.747i)15-s − 0.820·16-s + (0.594 − 1.02i)17-s + (0.411 + 0.821i)18-s + (0.787 + 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0901 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0901 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0901 - 0.995i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0901 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43461 + 1.57042i\)
\(L(\frac12)\) \(\approx\) \(1.43461 + 1.57042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 - 0.910i)T \)
7 \( 1 \)
good2 \( 1 - 1.29T + 2T^{2} \)
5 \( 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.589 + 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.61 - 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.45 + 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.43 - 5.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.14 + 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.92T + 31T^{2} \)
37 \( 1 + (-4.88 - 8.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.32 + 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.633T + 47T^{2} \)
53 \( 1 + (-1.11 + 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.21T + 59T^{2} \)
61 \( 1 + 9.65T + 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 + 3.27T + 71T^{2} \)
73 \( 1 + (0.519 - 0.898i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.00T + 79T^{2} \)
83 \( 1 + (3.65 - 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.02 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.46 - 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.48821551730534023237382316146, −10.47650047083386239517458775578, −9.641550227243442333397539020072, −8.593972548384648274162935700310, −7.66789827567543601822346528361, −6.70636510549851867125761489567, −5.44047804460374242969701638353, −4.20740557852997991756917767298, −3.48293567024641940690800576058, −2.72557196128189558922392932542, 1.01090539848272354918737622960, 3.04005386470932539281383387105, 3.96937868644088039684106776957, 4.90346400460244378518037750661, 5.88010784862385319216071319876, 7.37437557810211197822371395952, 8.185756538813103530401709285637, 8.926171342868556462523443746799, 9.622518079147039493278883979222, 11.27207412920473401932872721216

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