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Properties

Label 2-21e2-63.59-c1-0-25
Degree $2$
Conductor $441$
Sign $-0.0920 + 0.995i$
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  + (0.555 + 0.320i)2-s + (−0.175 + 1.72i)3-s + (−0.794 − 1.37i)4-s − 2.21·5-s + (−0.650 + 0.901i)6-s − 2.30i·8-s + (−2.93 − 0.605i)9-s + (−1.22 − 0.709i)10-s − 3.39i·11-s + (2.50 − 1.12i)12-s + (−1.56 − 0.901i)13-s + (0.388 − 3.80i)15-s + (−0.849 + 1.47i)16-s + (2.98 − 5.16i)17-s + (−1.43 − 1.27i)18-s + (−1.42 + 0.822i)19-s + ⋯
L(s)  = 1  + (0.392 + 0.226i)2-s + (−0.101 + 0.994i)3-s + (−0.397 − 0.687i)4-s − 0.988·5-s + (−0.265 + 0.367i)6-s − 0.813i·8-s + (−0.979 − 0.201i)9-s + (−0.388 − 0.224i)10-s − 1.02i·11-s + (0.724 − 0.325i)12-s + (−0.432 − 0.249i)13-s + (0.100 − 0.983i)15-s + (−0.212 + 0.367i)16-s + (0.723 − 1.25i)17-s + (−0.338 − 0.301i)18-s + (−0.326 + 0.188i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0920 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0920 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0920 + 0.995i$
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: \((2,\ 441,\ (\ :1/2),\ -0.0920 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418799 - 0.459311i\)
\(L(\frac12)\) \(\approx\) \(0.418799 - 0.459311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.175 - 1.72i)T \)
7 \( 1 \)
good2 \( 1 + (-0.555 - 0.320i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.21T + 5T^{2} \)
11 \( 1 + 3.39iT - 11T^{2} \)
13 \( 1 + (1.56 + 0.901i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.98 + 5.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.42 - 0.822i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.37iT - 23T^{2} \)
29 \( 1 + (-2.44 + 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (9.28 - 5.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.849 + 1.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.455 - 0.788i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.96 + 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.123 + 0.213i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.82 - 3.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.39 + 9.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.22 + 0.709i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.99 - 6.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (0.369 + 0.213i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.49 + 4.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.28 + 7.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.26 + 9.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.30 - 3.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.83331204057320386656160999575, −10.05602634440720067250846169576, −9.140087866003999036609958397257, −8.313460301512260491431914779431, −7.08666126310330231834353948024, −5.78139077127558205585989553973, −5.08353901255532648674963552287, −4.08178698022799817322808176060, −3.18664840487651189971980459867, −0.34479314126288309126594412402, 2.01045239303721220818018516916, 3.43221233574316024255357088469, 4.38816716225259815571534427823, 5.58997526850453259298517662889, 6.99661178456698657997456856767, 7.70906710587531367548176981461, 8.294979001833977833408060008801, 9.403932136375747845970928496205, 10.82545016804661638222484135306, 11.73874686417768158056465025717

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