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Properties

Label 2-21e2-63.5-c1-0-31
Degree $2$
Conductor $441$
Sign $-0.936 - 0.350i$
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  − 1.51i·2-s + (−0.811 − 1.53i)3-s − 0.280·4-s + (0.387 + 0.671i)5-s + (−2.31 + 1.22i)6-s − 2.59i·8-s + (−1.68 + 2.48i)9-s + (1.01 − 0.585i)10-s + (−3.32 − 1.92i)11-s + (0.227 + 0.429i)12-s + (−2.54 − 1.46i)13-s + (0.713 − 1.13i)15-s − 4.48·16-s + (−2.69 − 4.67i)17-s + (3.74 + 2.54i)18-s + (0.376 + 0.217i)19-s + ⋯
L(s)  = 1  − 1.06i·2-s + (−0.468 − 0.883i)3-s − 0.140·4-s + (0.173 + 0.300i)5-s + (−0.943 + 0.500i)6-s − 0.918i·8-s + (−0.561 + 0.827i)9-s + (0.320 − 0.185i)10-s + (−1.00 − 0.579i)11-s + (0.0656 + 0.123i)12-s + (−0.705 − 0.407i)13-s + (0.184 − 0.294i)15-s − 1.12·16-s + (−0.654 − 1.13i)17-s + (0.883 + 0.599i)18-s + (0.0863 + 0.0498i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.936 - 0.350i$
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: \((2,\ 441,\ (\ :1/2),\ -0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169886 + 0.938337i\)
\(L(\frac12)\) \(\approx\) \(0.169886 + 0.938337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.811 + 1.53i)T \)
7 \( 1 \)
good2 \( 1 + 1.51iT - 2T^{2} \)
5 \( 1 + (-0.387 - 0.671i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.32 + 1.92i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.54 + 1.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.376 - 0.217i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0482 - 0.0278i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.187 - 0.108i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.55iT - 31T^{2} \)
37 \( 1 + (-3.14 + 5.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.78 + 6.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.42 - 11.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.965T + 47T^{2} \)
53 \( 1 + (-6.46 + 3.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.12T + 59T^{2} \)
61 \( 1 - 3.48iT - 61T^{2} \)
67 \( 1 + 4.20T + 67T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (-7.05 + 4.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.96T + 79T^{2} \)
83 \( 1 + (4.31 + 7.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.82 + 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.24 + 0.716i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.80384556415025399703104112017, −10.16371266912736808653373539824, −8.954028317810410395575892632997, −7.67775440575434678749903320953, −6.96706744398619899586467761979, −5.89922418643128436121428229471, −4.76789150338757374196451347678, −2.95178901499660369923101562435, −2.31400149155128380771377675041, −0.59550883979910780313247642802, 2.42866458747154505695985043903, 4.24793809312214793875395123098, 5.12739530774236922357271366512, 5.89431271796215118650818814522, 6.88758231684304643012528489254, 7.87923808912959661729239842916, 8.848857466102727681888371120901, 9.756987748283020407858235813669, 10.68082267493160035079339585414, 11.44400100114570966893091833572

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