Properties

Label 2-21e2-441.142-c1-0-40
Degree $2$
Conductor $441$
Sign $0.687 + 0.726i$
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  + (−0.792 + 2.01i)2-s + (1.54 − 0.777i)3-s + (−1.98 − 1.83i)4-s + (−3.11 + 1.49i)5-s + (0.342 + 3.74i)6-s + (−0.809 − 2.51i)7-s + (1.37 − 0.660i)8-s + (1.79 − 2.40i)9-s + (−0.560 − 7.47i)10-s + (−2.48 − 3.11i)11-s + (−4.49 − 1.30i)12-s + (1.79 − 4.58i)13-s + (5.72 + 0.361i)14-s + (−3.65 + 4.74i)15-s + (−0.157 − 2.09i)16-s + (1.04 − 0.970i)17-s + ⋯
L(s)  = 1  + (−0.560 + 1.42i)2-s + (0.893 − 0.448i)3-s + (−0.990 − 0.918i)4-s + (−1.39 + 0.670i)5-s + (0.139 + 1.52i)6-s + (−0.305 − 0.952i)7-s + (0.485 − 0.233i)8-s + (0.597 − 0.801i)9-s + (−0.177 − 2.36i)10-s + (−0.749 − 0.940i)11-s + (−1.29 − 0.376i)12-s + (0.498 − 1.27i)13-s + (1.53 + 0.0966i)14-s + (−0.943 + 1.22i)15-s + (−0.0392 − 0.524i)16-s + (0.253 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.687 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509424 - 0.219243i\)
\(L(\frac12)\) \(\approx\) \(0.509424 - 0.219243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 + 0.777i)T \)
7 \( 1 + (0.809 + 2.51i)T \)
good2 \( 1 + (0.792 - 2.01i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (3.11 - 1.49i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (2.48 + 3.11i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.79 + 4.58i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-1.04 + 0.970i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (3.96 - 6.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.312 - 1.36i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.98 + 1.53i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-2.47 + 4.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.415 + 0.128i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (5.50 + 3.75i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (3.04 - 2.07i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-1.49 + 3.79i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-2.33 + 0.719i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.93 + 1.32i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-2.88 + 2.67i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (5.56 - 9.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.11 + 9.24i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (2.59 + 0.391i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (6.24 + 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.03 - 7.73i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-3.08 - 7.85i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (5.57 - 9.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70639644678970340967597322281, −9.971448867002854464935515054574, −8.550467434785894170628890748249, −7.913361608856123914859311867260, −7.68620647989690690838052102314, −6.71568024111430178833147294638, −5.75186821239782380600077883561, −3.89383988176869986749416835893, −3.17106696982061519633269967975, −0.36916657542948495354367681429, 1.91412478186110703107160864501, 2.97757098062116462076115656867, 4.08217953007981273781371360394, 4.81010737344232302232103104232, 6.94597852783785761057779526387, 8.208736903189542268767508328827, 8.797832183403524688319014521311, 9.313082597921396536373360023111, 10.34384481158015700138371513065, 11.26279414932987872519881323804

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