Properties

Label 2-21e2-63.5-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.140 - 0.990i$
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  + 1.86i·2-s + (1.72 + 0.174i)3-s − 1.49·4-s + (−1.25 − 2.17i)5-s + (−0.326 + 3.22i)6-s + 0.947i·8-s + (2.93 + 0.601i)9-s + (4.05 − 2.34i)10-s + (4.85 + 2.80i)11-s + (−2.57 − 0.260i)12-s + (−0.384 − 0.221i)13-s + (−1.78 − 3.95i)15-s − 4.75·16-s + (1.53 + 2.66i)17-s + (−1.12 + 5.49i)18-s + (2.22 + 1.28i)19-s + ⋯
L(s)  = 1  + 1.32i·2-s + (0.994 + 0.100i)3-s − 0.746·4-s + (−0.560 − 0.970i)5-s + (−0.133 + 1.31i)6-s + 0.335i·8-s + (0.979 + 0.200i)9-s + (1.28 − 0.740i)10-s + (1.46 + 0.845i)11-s + (−0.742 − 0.0751i)12-s + (−0.106 − 0.0615i)13-s + (−0.459 − 1.02i)15-s − 1.18·16-s + (0.373 + 0.646i)17-s + (−0.264 + 1.29i)18-s + (0.511 + 0.295i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.140 - 0.990i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.140 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26423 + 1.45661i\)
\(L(\frac12)\) \(\approx\) \(1.26423 + 1.45661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.174i)T \)
7 \( 1 \)
good2 \( 1 - 1.86iT - 2T^{2} \)
5 \( 1 + (1.25 + 2.17i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.85 - 2.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.384 + 0.221i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.53 - 2.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.22 - 1.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.83 - 3.94i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.71 + 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.14T + 47T^{2} \)
53 \( 1 + (-4.20 + 2.42i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.30T + 59T^{2} \)
61 \( 1 + 8.55iT - 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 - 2.95iT - 71T^{2} \)
73 \( 1 + (7.37 - 4.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.574T + 79T^{2} \)
83 \( 1 + (4.23 + 7.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.78 - 6.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.22 + 1.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.67440449940674884103350678516, −9.928885329819561549807110677764, −9.275733685088843034714372528553, −8.321112260829252908758998580147, −7.84718730665963856154193050142, −6.94243952216263650250860703759, −5.83411533229806658086434449960, −4.53239649015311366868373886972, −3.85677872047998405200763087403, −1.80559318869493659802777653488, 1.36608007326560906756154640737, 2.88549496415769422582627603290, 3.39492880997980871258763134369, 4.37129407201147629212680142187, 6.50436819516078322820997034098, 7.17452332044060973734563736289, 8.417105482924971878822333927517, 9.259232291920838906688633538749, 10.12377821535296417750699897688, 10.88425007297867593226190841671

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