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Properties

Label 2-21e2-7.2-c3-0-30
Degree $2$
Conductor $441$
Sign $0.991 + 0.126i$
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  + (−2.63 + 4.56i)2-s + (−9.91 − 17.1i)4-s + (5.27 − 9.13i)5-s + 62.3·8-s + (27.8 + 48.1i)10-s + (17.3 + 30.0i)11-s − 37.2·13-s + (−85.2 + 147. i)16-s + (−5.27 − 9.13i)17-s + (29.2 − 50.7i)19-s − 209.·20-s − 183.·22-s + (−62.6 + 108. i)23-s + (6.85 + 11.8i)25-s + (98.3 − 170. i)26-s + ⋯
L(s)  = 1  + (−0.932 + 1.61i)2-s + (−1.23 − 2.14i)4-s + (0.471 − 0.817i)5-s + 2.75·8-s + (0.879 + 1.52i)10-s + (0.476 + 0.824i)11-s − 0.795·13-s + (−1.33 + 2.30i)16-s + (−0.0752 − 0.130i)17-s + (0.353 − 0.612i)19-s − 2.33·20-s − 1.77·22-s + (−0.568 + 0.984i)23-s + (0.0548 + 0.0949i)25-s + (0.742 − 1.28i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8083499792\)
\(L(\frac12)\) \(\approx\) \(0.8083499792\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.63 - 4.56i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-5.27 + 9.13i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-17.3 - 30.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 37.2T + 2.19e3T^{2} \)
17 \( 1 + (5.27 + 9.13i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-29.2 + 50.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (62.6 - 108. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 35.4T + 2.43e4T^{2} \)
31 \( 1 + (145. + 252. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-129. + 225. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 338.T + 6.89e4T^{2} \)
43 \( 1 - 6.80T + 7.95e4T^{2} \)
47 \( 1 + (-125. + 217. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (268. + 464. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (17.9 + 31.0i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (28.8 - 50.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (240. + 417. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 363.T + 3.57e5T^{2} \)
73 \( 1 + (290. + 503. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-346. + 600. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + (176. - 305. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.44e3T + 9.12e5T^{2} \)
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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

−10.04960911284394609783686703099, −9.394918249582309795350555948528, −9.007171056764296141528684194400, −7.70887290058268337892042066929, −7.25010443349320313655467647837, −6.06692356030992509089717607215, −5.27831217177298894793606939471, −4.38932228561993039311415171594, −1.81414456876622575841962064649, −0.41563477382527532338172114273, 1.14313618393196912280212985987, 2.44659918695036171573150551731, 3.21678054004419363902619678191, 4.41967751051134351224267921482, 6.09899216010792745419136971054, 7.34203235972544338880367265736, 8.374055648574456712468271951955, 9.186262989778482649681558035077, 10.09041022073045276691899204941, 10.58744312484837832972635255034

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