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Properties

Label 2-21e2-7.4-c3-0-39
Degree $2$
Conductor $441$
Sign $0.266 + 0.963i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.17 + 3.77i)2-s + (−5.5 + 9.52i)4-s + (−4.35 − 7.54i)5-s − 13.0·8-s + (19 − 32.9i)10-s + (−21.7 + 37.7i)11-s − 82·13-s + (15.4 + 26.8i)16-s + (39.2 − 67.9i)17-s + (−10 − 17.3i)19-s + 95.8·20-s − 190·22-s + (−65.3 − 113. i)23-s + (24.5 − 42.4i)25-s + (−178. − 309. i)26-s + ⋯
L(s)  = 1  + (0.770 + 1.33i)2-s + (−0.687 + 1.19i)4-s + (−0.389 − 0.675i)5-s − 0.577·8-s + (0.600 − 1.04i)10-s + (−0.597 + 1.03i)11-s − 1.74·13-s + (0.242 + 0.419i)16-s + (0.559 − 0.969i)17-s + (−0.120 − 0.209i)19-s + 1.07·20-s − 1.84·22-s + (−0.592 − 1.02i)23-s + (0.196 − 0.339i)25-s + (−1.34 − 2.33i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3739709445\)
\(L(\frac12)\) \(\approx\) \(0.3739709445\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + (-2.17 - 3.77i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (4.35 + 7.54i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (21.7 - 37.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 82T + 2.19e3T^{2} \)
17 \( 1 + (-39.2 + 67.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (10 + 17.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (65.3 + 113. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 244.T + 2.43e4T^{2} \)
31 \( 1 + (-78 + 135. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (93 + 161. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 + (235. + 407. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (78.4 - 135. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (78.4 - 135. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-395 - 684. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-22 + 38.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 444.T + 3.57e5T^{2} \)
73 \( 1 + (-63 + 109. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-356 - 616. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.46e3T + 5.71e5T^{2} \)
89 \( 1 + (-727. - 1.26e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 798T + 9.12e5T^{2} \)
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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

−10.32852246742933276479443904316, −9.484369971712076835310107905903, −8.278347179850254067885593428795, −7.46702841948836548474665175444, −6.97308116449662541965760247458, −5.53931503650434187631812992955, −4.87672110078471818482797742010, −4.17172999665541428182393437925, −2.38825225152169387291481784431, −0.086813925384842087805861367126, 1.73218005608949624100579720343, 2.98643933421661557396188505889, 3.60201851758324433434076910245, 4.89536374993269176840846945965, 5.79492471348128246791146930413, 7.27497964094521572076266265904, 8.079916877076192589918186019439, 9.596775226499587351360574284869, 10.28958646399863773874953571178, 11.06260428740643163733694573043

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