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Properties

Label 2-21e2-49.44-c1-0-16
Degree $2$
Conductor $441$
Sign $-0.690 + 0.723i$
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  + (−1.78 + 0.269i)2-s + (1.20 − 0.372i)4-s + (0.264 − 3.52i)5-s + (0.675 − 2.55i)7-s + (1.20 − 0.577i)8-s + (0.477 + 6.37i)10-s + (0.164 − 0.419i)11-s + (−0.882 − 1.10i)13-s + (−0.517 + 4.75i)14-s + (−4.07 + 2.77i)16-s + (−4.25 + 3.94i)17-s + (0.677 − 1.17i)19-s + (−0.993 − 4.35i)20-s + (−0.181 + 0.794i)22-s + (−0.724 − 0.672i)23-s + ⋯
L(s)  = 1  + (−1.26 + 0.190i)2-s + (0.603 − 0.186i)4-s + (0.118 − 1.57i)5-s + (0.255 − 0.966i)7-s + (0.424 − 0.204i)8-s + (0.151 + 2.01i)10-s + (0.0496 − 0.126i)11-s + (−0.244 − 0.306i)13-s + (−0.138 + 1.26i)14-s + (−1.01 + 0.694i)16-s + (−1.03 + 0.957i)17-s + (0.155 − 0.269i)19-s + (−0.222 − 0.973i)20-s + (−0.0386 + 0.169i)22-s + (−0.151 − 0.140i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.690 + 0.723i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.208878 - 0.488098i\)
\(L(\frac12)\) \(\approx\) \(0.208878 - 0.488098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.675 + 2.55i)T \)
good2 \( 1 + (1.78 - 0.269i)T + (1.91 - 0.589i)T^{2} \)
5 \( 1 + (-0.264 + 3.52i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-0.164 + 0.419i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (0.882 + 1.10i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (4.25 - 3.94i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-0.677 + 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.724 + 0.672i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (0.913 + 4.00i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-1.11 - 1.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.68 - 0.827i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-7.49 + 3.60i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (4.88 + 2.35i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.01 - 0.152i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (10.0 - 3.11i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.382 + 5.10i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-3.43 - 1.06i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (7.60 + 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.56 - 11.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.27 + 0.192i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (3.99 - 6.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.09 + 6.38i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.13 - 2.88i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−10.52319577722453412935092023425, −9.687762072949752831824409652391, −8.898176199320423969480271726994, −8.230639754984345566166784181588, −7.51734983729160646599981624273, −6.29592579731719583114045690738, −4.84300412232396685304411186190, −4.11244440218263066822226789232, −1.69690319780022434523454237031, −0.51219466812085143260065875912, 2.01816080146458604401223719868, 2.91999411487272662953324881554, 4.72187553705283590396723690586, 6.14471345674614534629996631919, 7.08367544341126442860862733112, 7.84871763352974691050524807566, 8.974867786203070659401742130276, 9.599656069128422873899509424879, 10.46010486400239908040027506812, 11.29720098031641842179487046242

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