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Properties

Label 2-21e2-63.41-c1-0-17
Degree $2$
Conductor $441$
Sign $0.842 - 0.538i$
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  + (1.80 − 1.04i)2-s + (0.886 + 1.48i)3-s + (1.17 − 2.03i)4-s + (−1.65 + 2.86i)5-s + (3.15 + 1.76i)6-s − 0.717i·8-s + (−1.42 + 2.63i)9-s + 6.88i·10-s + (2.30 − 1.33i)11-s + (4.06 − 0.0560i)12-s + (−2.11 − 1.21i)13-s + (−5.72 + 0.0790i)15-s + (1.59 + 2.76i)16-s + 7.18·17-s + (0.172 + 6.25i)18-s − 4.90i·19-s + ⋯
L(s)  = 1  + (1.27 − 0.736i)2-s + (0.511 + 0.859i)3-s + (0.586 − 1.01i)4-s + (−0.738 + 1.27i)5-s + (1.28 + 0.719i)6-s − 0.253i·8-s + (−0.475 + 0.879i)9-s + 2.17i·10-s + (0.694 − 0.401i)11-s + (1.17 − 0.0161i)12-s + (−0.585 − 0.338i)13-s + (−1.47 + 0.0203i)15-s + (0.399 + 0.691i)16-s + 1.74·17-s + (0.0406 + 1.47i)18-s − 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68461 + 0.784389i\)
\(L(\frac12)\) \(\approx\) \(2.68461 + 0.784389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.886 - 1.48i)T \)
7 \( 1 \)
good2 \( 1 + (-1.80 + 1.04i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.65 - 2.86i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.30 + 1.33i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.11 + 1.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 + 4.90iT - 19T^{2} \)
23 \( 1 + (4.32 + 2.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.50 + 3.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.30 + 1.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.68T + 37T^{2} \)
41 \( 1 + (0.553 - 0.958i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.93 - 5.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.44 + 4.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (-2.56 + 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.16 - 7.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.07iT - 71T^{2} \)
73 \( 1 - 8.01iT - 73T^{2} \)
79 \( 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.04 + 1.80i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.08T + 89T^{2} \)
97 \( 1 + (-9.47 + 5.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.45477066793316472381156121181, −10.45353122720513979905830541012, −9.911903714149669798441684880144, −8.446704516494112517676755625725, −7.52329340478475720539015236531, −6.22083718920610739589049159405, −5.06990818688719358834138325479, −4.01396130008388109236634206892, −3.29079831567307554931585639495, −2.56626967498448655096772980219, 1.34604791251907253502975860058, 3.39171406132407008358635602842, 4.22720200408278582873645604841, 5.31376104405271447156063005416, 6.21952847579026054138395699548, 7.42728244937920232818020552507, 7.85240327106198788743018370852, 8.939089380932416570549794696519, 9.922582267100751828732167240323, 11.98958684095287657175234988153

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