Properties

Label 2-21e2-441.337-c1-0-30
Degree $2$
Conductor $441$
Sign $0.505 - 0.862i$
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  + (1.89 + 0.284i)2-s + (1.05 + 1.37i)3-s + (1.58 + 0.487i)4-s + (−0.509 + 0.347i)5-s + (1.59 + 2.90i)6-s + (2.55 + 0.698i)7-s + (−0.595 − 0.286i)8-s + (−0.790 + 2.89i)9-s + (−1.06 + 0.511i)10-s + (1.83 + 0.277i)11-s + (0.989 + 2.68i)12-s + (−1.35 − 3.44i)13-s + (4.62 + 2.04i)14-s + (−1.01 − 0.336i)15-s + (−3.77 − 2.57i)16-s + (0.108 − 0.476i)17-s + ⋯
L(s)  = 1  + (1.33 + 0.201i)2-s + (0.606 + 0.794i)3-s + (0.790 + 0.243i)4-s + (−0.227 + 0.155i)5-s + (0.650 + 1.18i)6-s + (0.964 + 0.263i)7-s + (−0.210 − 0.101i)8-s + (−0.263 + 0.964i)9-s + (−0.335 + 0.161i)10-s + (0.554 + 0.0835i)11-s + (0.285 + 0.776i)12-s + (−0.375 − 0.956i)13-s + (1.23 + 0.547i)14-s + (−0.261 − 0.0868i)15-s + (−0.944 − 0.643i)16-s + (0.0263 − 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75287 + 1.57712i\)
\(L(\frac12)\) \(\approx\) \(2.75287 + 1.57712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.05 - 1.37i)T \)
7 \( 1 + (-2.55 - 0.698i)T \)
good2 \( 1 + (-1.89 - 0.284i)T + (1.91 + 0.589i)T^{2} \)
5 \( 1 + (0.509 - 0.347i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (-1.83 - 0.277i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (1.35 + 3.44i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (-0.108 + 0.476i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 4.81T + 19T^{2} \)
23 \( 1 + (-2.49 - 0.770i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (-9.29 + 2.86i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-3.05 + 5.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.66 + 7.29i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (7.15 - 4.87i)T + (14.9 - 38.1i)T^{2} \)
43 \( 1 + (5.57 + 3.80i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (0.579 + 0.0872i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-1.57 - 6.91i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-0.915 + 12.2i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (10.4 - 3.23i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-3.05 + 5.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.553 + 2.42i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.39 + 1.75i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (0.767 + 1.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.22 - 5.67i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-5.10 - 6.40i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-3.74 - 6.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.42917982059303646148186147121, −10.52993433317813532517797026369, −9.453830863782377268747809559668, −8.480684511990823256597504041243, −7.61121060176291239299350614491, −6.26918563428556128798422628365, −5.15335220000342769465593629621, −4.53004496186981723969518100853, −3.54528209397526034152205147527, −2.43765065669704424474817513735, 1.66272178645795322483317364587, 2.91292743793275486334275775391, 4.19048372279343481530052383581, 4.83627580853060722074127093459, 6.35096880799129856623251315997, 6.93891051485509171571850551222, 8.402826365528390908541775449970, 8.717533237608990416100635333791, 10.28753109976449922168448656937, 11.61728870201664510910898078992

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