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Properties

Label 2-21e2-63.58-c1-0-29
Degree $2$
Conductor $441$
Sign $-0.210 + 0.977i$
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  + 1.69·2-s + (−1.29 + 1.15i)3-s + 0.888·4-s + (−1.79 − 3.10i)5-s + (−2.19 + 1.95i)6-s − 1.88·8-s + (0.349 − 2.97i)9-s + (−3.04 − 5.28i)10-s + (1.40 − 2.43i)11-s + (−1.15 + 1.02i)12-s + (0.5 − 0.866i)13-s + (5.89 + 1.95i)15-s − 4.98·16-s + (−2.05 − 3.56i)17-s + (0.594 − 5.06i)18-s + (−0.444 + 0.769i)19-s + ⋯
L(s)  = 1  + 1.20·2-s + (−0.747 + 0.664i)3-s + 0.444·4-s + (−0.802 − 1.38i)5-s + (−0.897 + 0.798i)6-s − 0.667·8-s + (0.116 − 0.993i)9-s + (−0.964 − 1.67i)10-s + (0.423 − 0.733i)11-s + (−0.332 + 0.295i)12-s + (0.138 − 0.240i)13-s + (1.52 + 0.505i)15-s − 1.24·16-s + (−0.498 − 0.863i)17-s + (0.140 − 1.19i)18-s + (−0.101 + 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.210 + 0.977i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.210 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.719428 - 0.891146i\)
\(L(\frac12)\) \(\approx\) \(0.719428 - 0.891146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.29 - 1.15i)T \)
7 \( 1 \)
good2 \( 1 - 1.69T + 2T^{2} \)
5 \( 1 + (1.79 + 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.05 + 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.444 - 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.93 + 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.98T + 31T^{2} \)
37 \( 1 + (2.38 - 4.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.70 - 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 + (-0.0618 - 0.107i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.87T + 59T^{2} \)
61 \( 1 + 3.87T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + (5.32 + 9.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + (2.05 + 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.80 + 8.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.37447965965307122938467019974, −10.06870572235943975539236036878, −8.929943146717076140009410255695, −8.399303079785936027718992507488, −6.68168170424727991975189245820, −5.73951986305941898485664444808, −4.79666396377839468859946073020, −4.34244755339517652840843750407, −3.30205228751117924941101815140, −0.53400593059500691858953867677, 2.27842928313334343828653154111, 3.65017604043491863373875568716, 4.49231979571777572635090336742, 5.79855275105721094286133080386, 6.61805188720198292615774379863, 7.19882610087466827873696826693, 8.341409784588991914384703087442, 9.918935202818988036004435193194, 10.92928096777734709969072281664, 11.64755723541431522188024931047

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