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Properties

Label 2-21e2-441.437-c1-0-32
Degree $2$
Conductor $441$
Sign $0.379 - 0.925i$
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  + (2.52 + 0.189i)2-s + (−0.882 + 1.49i)3-s + (4.36 + 0.657i)4-s + (−0.486 + 2.13i)5-s + (−2.51 + 3.59i)6-s + (1.52 − 2.16i)7-s + (5.95 + 1.35i)8-s + (−1.44 − 2.63i)9-s + (−1.63 + 5.29i)10-s + (0.444 + 0.922i)11-s + (−4.83 + 5.92i)12-s + (1.89 + 0.142i)13-s + (4.25 − 5.17i)14-s + (−2.74 − 2.60i)15-s + (6.35 + 1.96i)16-s + (−6.21 + 0.937i)17-s + ⋯
L(s)  = 1  + (1.78 + 0.133i)2-s + (−0.509 + 0.860i)3-s + (2.18 + 0.328i)4-s + (−0.217 + 0.953i)5-s + (−1.02 + 1.46i)6-s + (0.575 − 0.818i)7-s + (2.10 + 0.480i)8-s + (−0.480 − 0.876i)9-s + (−0.516 + 1.67i)10-s + (0.133 + 0.278i)11-s + (−1.39 + 1.71i)12-s + (0.525 + 0.0394i)13-s + (1.13 − 1.38i)14-s + (−0.709 − 0.672i)15-s + (1.58 + 0.490i)16-s + (−1.50 + 0.227i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.379 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69904 + 1.81072i\)
\(L(\frac12)\) \(\approx\) \(2.69904 + 1.81072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.882 - 1.49i)T \)
7 \( 1 + (-1.52 + 2.16i)T \)
good2 \( 1 + (-2.52 - 0.189i)T + (1.97 + 0.298i)T^{2} \)
5 \( 1 + (0.486 - 2.13i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-0.444 - 0.922i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.89 - 0.142i)T + (12.8 + 1.93i)T^{2} \)
17 \( 1 + (6.21 - 0.937i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (1.54 - 0.891i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.10 + 1.68i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (1.02 + 0.404i)T + (21.2 + 19.7i)T^{2} \)
31 \( 1 + (-9.15 + 5.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.78 + 7.10i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (3.65 + 3.39i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (3.77 - 3.50i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (-0.598 + 7.99i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (-7.46 + 2.92i)T + (38.8 - 36.0i)T^{2} \)
59 \( 1 + (4.58 - 4.25i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (0.860 + 5.71i)T + (-58.2 + 17.9i)T^{2} \)
67 \( 1 + (-7.17 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.496 - 0.396i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-5.25 - 7.70i)T + (-26.6 + 67.9i)T^{2} \)
79 \( 1 + (7.99 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.800 - 10.6i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (0.453 + 6.05i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (8.99 - 5.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.20778745735129284270305316847, −10.94629727559622833466165427126, −10.00787723774641712288157638056, −8.353506166225140248151866048097, −6.92929958317593230621273342724, −6.49427955836753544996167770333, −5.37364872968647556963591981307, −4.18583113052077280858444990601, −3.96059813040452041226844162392, −2.50107274114046599324798156892, 1.58974899408035182726156012737, 2.80612685260887881385381224863, 4.51733649724791539686520283711, 5.00228183982468279170315556452, 6.07056170444536056225961075859, 6.67443784095559115958062315555, 8.078609629734127654200888820053, 8.833820009424233370255492837109, 10.73510973022144740990383704098, 11.51752380887431174557146201073

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