Properties

Label 2-21e2-441.110-c1-0-23
Degree $2$
Conductor $441$
Sign $0.587 - 0.809i$
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  + (0.875 − 0.0655i)2-s + (1.63 + 0.563i)3-s + (−1.21 + 0.183i)4-s + (0.193 + 0.848i)5-s + (1.47 + 0.386i)6-s + (2.18 + 1.49i)7-s + (−2.76 + 0.630i)8-s + (2.36 + 1.84i)9-s + (0.225 + 0.729i)10-s + (−0.798 + 1.65i)11-s + (−2.09 − 0.385i)12-s + (0.536 − 0.0401i)13-s + (2.00 + 1.16i)14-s + (−0.161 + 1.49i)15-s + (−0.0270 + 0.00833i)16-s + (−0.957 − 0.144i)17-s + ⋯
L(s)  = 1  + (0.618 − 0.0463i)2-s + (0.945 + 0.325i)3-s + (−0.608 + 0.0916i)4-s + (0.0866 + 0.379i)5-s + (0.600 + 0.157i)6-s + (0.825 + 0.564i)7-s + (−0.977 + 0.223i)8-s + (0.787 + 0.615i)9-s + (0.0712 + 0.230i)10-s + (−0.240 + 0.500i)11-s + (−0.604 − 0.111i)12-s + (0.148 − 0.0111i)13-s + (0.536 + 0.311i)14-s + (−0.0416 + 0.387i)15-s + (−0.00675 + 0.00208i)16-s + (−0.232 − 0.0350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.587 - 0.809i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.587 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02294 + 1.03176i\)
\(L(\frac12)\) \(\approx\) \(2.02294 + 1.03176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 - 0.563i)T \)
7 \( 1 + (-2.18 - 1.49i)T \)
good2 \( 1 + (-0.875 + 0.0655i)T + (1.97 - 0.298i)T^{2} \)
5 \( 1 + (-0.193 - 0.848i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (0.798 - 1.65i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.536 + 0.0401i)T + (12.8 - 1.93i)T^{2} \)
17 \( 1 + (0.957 + 0.144i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (2.82 + 1.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.99 + 3.18i)T + (5.11 - 22.4i)T^{2} \)
29 \( 1 + (2.42 - 0.952i)T + (21.2 - 19.7i)T^{2} \)
31 \( 1 + (-3.04 - 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.43 + 3.66i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-2.25 + 2.08i)T + (3.06 - 40.8i)T^{2} \)
43 \( 1 + (3.25 + 3.01i)T + (3.21 + 42.8i)T^{2} \)
47 \( 1 + (-0.0375 - 0.500i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (10.0 + 3.95i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (7.02 + 6.51i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-1.22 + 8.14i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (-4.48 + 7.77i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.1 - 8.87i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.26 + 6.26i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (1.55 + 2.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0843 - 1.12i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (0.00186 - 0.0248i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (-12.7 - 7.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.20540935661573993663905466246, −10.36035261575186749569546010040, −9.206283506178550041593039372626, −8.680358166076649649509266625464, −7.80279978500601038908300798097, −6.54408560864848350730798931613, −5.06781500457620310306600442319, −4.53041803839602383954847029619, −3.25877736656745029088392995762, −2.21315900621809977505097022020, 1.28255967713563250960105458993, 3.04003639058123133715775192727, 4.14049504759902442555801836499, 4.95614202037125698267055947346, 6.18777436043591511765648584916, 7.43834549451628804104631255302, 8.380548485672865679216132721822, 8.951307863896994923657647592630, 9.937419892641774918218901967857, 11.04236830601095691395222147047

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