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Properties

Label 2-21e2-49.29-c1-0-11
Degree $2$
Conductor $441$
Sign $-0.0198 - 0.999i$
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  + (2.06 + 0.996i)2-s + (2.04 + 2.56i)4-s + (0.349 + 1.53i)5-s + (−0.354 + 2.62i)7-s + (0.653 + 2.86i)8-s + (−0.803 + 3.51i)10-s + (−4.49 − 2.16i)11-s + (1.84 + 0.887i)13-s + (−3.34 + 5.07i)14-s + (−0.0415 + 0.182i)16-s + (1.20 − 1.51i)17-s + 6.63·19-s + (−3.21 + 4.02i)20-s + (−7.14 − 8.95i)22-s + (−1.01 − 1.26i)23-s + ⋯
L(s)  = 1  + (1.46 + 0.704i)2-s + (1.02 + 1.28i)4-s + (0.156 + 0.684i)5-s + (−0.133 + 0.990i)7-s + (0.230 + 1.01i)8-s + (−0.253 + 1.11i)10-s + (−1.35 − 0.652i)11-s + (0.510 + 0.246i)13-s + (−0.894 + 1.35i)14-s + (−0.0103 + 0.0455i)16-s + (0.292 − 0.367i)17-s + 1.52·19-s + (−0.717 + 0.900i)20-s + (−1.52 − 1.90i)22-s + (−0.211 − 0.264i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0198 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0198 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0198 - 0.999i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.0198 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06476 + 2.10619i\)
\(L(\frac12)\) \(\approx\) \(2.06476 + 2.10619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.354 - 2.62i)T \)
good2 \( 1 + (-2.06 - 0.996i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (-0.349 - 1.53i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (4.49 + 2.16i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.84 - 0.887i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-1.20 + 1.51i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 - 6.63T + 19T^{2} \)
23 \( 1 + (1.01 + 1.26i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.96 + 4.97i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 + (3.87 - 4.86i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-1.76 - 7.72i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.23 + 5.40i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.138 - 0.0665i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.44 + 3.06i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (2.33 - 10.2i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-5.14 + 6.45i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 + (5.76 + 7.23i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-4.80 + 2.31i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 8.93T + 79T^{2} \)
83 \( 1 + (-6.94 + 3.34i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.997 - 0.480i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−11.68724138653948350956531508765, −10.68587914098502496941686175013, −9.569307213184217558901809318781, −8.282738707529876660198562306241, −7.38805201607086530184707324137, −6.33878401787697774118198375248, −5.64143770914783439106205336840, −4.88898084455297911602941123142, −3.31909802390267133561776495915, −2.69618723593729392966445521247, 1.41982768340041969356806069929, 2.98821066696735718902844401046, 3.96048853329132930325967247512, 5.07489579196237788040017824292, 5.55272427650247196540472373361, 7.03134513351269885306334981920, 7.977316726053669902249740925388, 9.368813794834337991146564130042, 10.49368228228709483687596573600, 10.86335952524204692735324739481

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