Properties

Label 2-21e2-49.16-c1-0-16
Degree $2$
Conductor $441$
Sign $0.999 + 0.0426i$
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  + (0.354 + 0.902i)2-s + (0.777 − 0.720i)4-s + (0.605 − 0.413i)5-s + (−0.310 − 2.62i)7-s + (2.67 + 1.28i)8-s + (0.587 + 0.400i)10-s + (−1.30 − 0.197i)11-s + (−0.696 + 0.873i)13-s + (2.26 − 1.21i)14-s + (−0.0565 + 0.754i)16-s + (5.89 − 1.81i)17-s + (2.99 − 5.18i)19-s + (0.172 − 0.757i)20-s + (−0.285 − 1.25i)22-s + (−5.13 − 1.58i)23-s + ⋯
L(s)  = 1  + (0.250 + 0.638i)2-s + (0.388 − 0.360i)4-s + (0.270 − 0.184i)5-s + (−0.117 − 0.993i)7-s + (0.945 + 0.455i)8-s + (0.185 + 0.126i)10-s + (−0.394 − 0.0594i)11-s + (−0.193 + 0.242i)13-s + (0.604 − 0.323i)14-s + (−0.0141 + 0.188i)16-s + (1.43 − 0.441i)17-s + (0.687 − 1.19i)19-s + (0.0386 − 0.169i)20-s + (−0.0608 − 0.266i)22-s + (−1.07 − 0.330i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0426i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.999 + 0.0426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89683 - 0.0404533i\)
\(L(\frac12)\) \(\approx\) \(1.89683 - 0.0404533i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.310 + 2.62i)T \)
good2 \( 1 + (-0.354 - 0.902i)T + (-1.46 + 1.36i)T^{2} \)
5 \( 1 + (-0.605 + 0.413i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (1.30 + 0.197i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (0.696 - 0.873i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-5.89 + 1.81i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-2.99 + 5.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.13 + 1.58i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (0.529 - 2.32i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-3.15 - 5.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.10 - 3.81i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-0.798 - 0.384i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (5.17 - 2.49i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.440 - 1.12i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (6.99 - 6.48i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (0.392 + 0.267i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (2.56 + 2.38i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (2.53 + 4.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.06 - 9.05i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.19 + 3.05i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (-0.509 + 0.882i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.13 + 5.18i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (3.59 - 0.542i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−11.03970167138945475649467284218, −10.16916186340924761759264010229, −9.515316183657687699196674575719, −8.001125588886204466016899382215, −7.34369841856596172243227281854, −6.48897021754480116100664531252, −5.41327615033545568952965611799, −4.61251154890542053990284353346, −3.04616718481964235328940460701, −1.29914899273461256373369191812, 1.86530331384124914438140395489, 2.90636548320837123034850301147, 3.94850541855819642282845631207, 5.49306619810397421437799945725, 6.22484398769750451822262111810, 7.72018002507016094507964337692, 8.146535333488152477147358600545, 9.824101081430131530497131559929, 10.08151545291659266573665761539, 11.33515867621610663531160212727

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