Properties

Label 2-21e2-21.17-c1-0-3
Degree $2$
Conductor $441$
Sign $-0.462 - 0.886i$
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  + (−0.358 + 0.207i)2-s + (−0.914 + 1.58i)4-s + (1.46 + 2.53i)5-s − 1.58i·8-s + (−1.05 − 0.606i)10-s + (4.18 + 2.41i)11-s − 2.93i·13-s + (−1.49 − 2.59i)16-s + (−3.53 + 6.12i)17-s + (−5.07 + 2.93i)19-s − 5.35·20-s − 2·22-s + (1.73 − i)23-s + (−1.79 + 3.10i)25-s + (0.606 + 1.05i)26-s + ⋯
L(s)  = 1  + (−0.253 + 0.146i)2-s + (−0.457 + 0.791i)4-s + (0.655 + 1.13i)5-s − 0.560i·8-s + (−0.332 − 0.191i)10-s + (1.26 + 0.727i)11-s − 0.812i·13-s + (−0.374 − 0.649i)16-s + (−0.857 + 1.48i)17-s + (−1.16 + 0.672i)19-s − 1.19·20-s − 0.426·22-s + (0.361 − 0.208i)23-s + (−0.358 + 0.621i)25-s + (0.119 + 0.206i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.462 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589381 + 0.972304i\)
\(L(\frac12)\) \(\approx\) \(0.589381 + 0.972304i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (0.358 - 0.207i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.46 - 2.53i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.18 - 2.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + (3.53 - 6.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.07 - 2.93i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + (5.07 + 2.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.70 - 4.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 - 4.48T + 43T^{2} \)
47 \( 1 + (-2.93 - 5.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.12 + 3.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.93 + 5.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.05 - 0.606i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.24 + 7.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.828 + 1.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (5.60 + 9.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.07iT - 97T^{2} \)
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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

−11.22879736680530777051515374524, −10.44812771320595618083206021564, −9.601825264177506138373908796044, −8.711169748258876767827426900381, −7.77278611897289496159593466822, −6.70189816540234651160349704120, −6.14408517421830221248132448322, −4.39230686741291378255483478267, −3.48877780656809221776394884904, −2.08536823957015810192403323028, 0.817435497941986455493466828646, 2.09119416216593469476841220880, 4.20175228854807131295539411652, 5.00498950744690053462465164552, 6.00044826500447758174708338709, 6.94025150909168473493777404697, 8.731853538563880585840083935235, 9.055159077324915474100747359452, 9.542130590926080234903550636847, 10.90525934838017594978547447975

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