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Properties

Label 2-21e2-7.2-c3-0-26
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  + (1.13 − 1.97i)2-s + (1.41 + 2.44i)4-s + (−2.27 + 3.94i)5-s + 24.6·8-s + (5.17 + 8.96i)10-s + (−20.3 − 35.2i)11-s + 53.2·13-s + (16.7 − 28.9i)16-s + (2.27 + 3.94i)17-s + (−61.2 + 106. i)19-s − 12.8·20-s − 92.7·22-s + (65.6 − 113. i)23-s + (52.1 + 90.3i)25-s + (60.6 − 105. i)26-s + ⋯
L(s)  = 1  + (0.402 − 0.696i)2-s + (0.176 + 0.305i)4-s + (−0.203 + 0.352i)5-s + 1.08·8-s + (0.163 + 0.283i)10-s + (−0.558 − 0.967i)11-s + 1.13·13-s + (0.261 − 0.452i)16-s + (0.0324 + 0.0562i)17-s + (−0.740 + 1.28i)19-s − 0.143·20-s − 0.898·22-s + (0.595 − 1.03i)23-s + (0.417 + 0.722i)25-s + (0.457 − 0.792i)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

\(L(2)\) \(\approx\) \(2.737745716\)
\(L(\frac12)\) \(\approx\) \(2.737745716\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.13 + 1.97i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (2.27 - 3.94i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (20.3 + 35.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 53.2T + 2.19e3T^{2} \)
17 \( 1 + (-2.27 - 3.94i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (61.2 - 106. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-65.6 + 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 216.T + 2.43e4T^{2} \)
31 \( 1 + (-125. - 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (5.94 - 10.3i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 111.T + 6.89e4T^{2} \)
43 \( 1 - 369.T + 7.95e4T^{2} \)
47 \( 1 + (131. - 227. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (283. + 491. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-419. - 727. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-242. + 420. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-166. - 288. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 590.T + 3.57e5T^{2} \)
73 \( 1 + (245. + 424. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (60.8 - 105. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 609.T + 5.71e5T^{2} \)
89 \( 1 + (-359. + 622. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 637.T + 9.12e5T^{2} \)
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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

−10.75727860500200315673724851696, −10.33050253355976972248021500549, −8.620926516051447965672343158808, −8.150429222369500526780073658588, −6.91160243297082169381731331586, −5.95781937289982003910301433886, −4.57679872110058763843848078805, −3.50068073048632801504316173733, −2.74207287823686732510167972984, −1.19314018676567476159778557209, 0.952161377659516411008576707828, 2.46499507995740152647152630667, 4.25058211168099495238479215315, 4.94652375154764174859598021832, 6.05575762583689036866682641091, 6.88217638156899595831087168303, 7.79457273251849276839566237018, 8.754342331719304119437162784433, 9.872094858744564351142329457913, 10.75702013305947847779750788496

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