Properties

Label 2-21e2-147.104-c1-0-13
Degree $2$
Conductor $441$
Sign $0.701 - 0.713i$
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  + (1.42 + 1.13i)2-s + (0.295 + 1.29i)4-s + (−0.0720 − 0.0346i)5-s + (2.51 − 0.832i)7-s + (0.533 − 1.10i)8-s + (−0.0632 − 0.131i)10-s + (−0.0489 − 0.0390i)11-s + (3.06 + 2.44i)13-s + (4.52 + 1.66i)14-s + (4.40 − 2.12i)16-s + (−0.399 + 1.75i)17-s + 5.46i·19-s + (0.0235 − 0.103i)20-s + (−0.0254 − 0.111i)22-s + (−4.14 + 0.945i)23-s + ⋯
L(s)  = 1  + (1.00 + 0.804i)2-s + (0.147 + 0.646i)4-s + (−0.0322 − 0.0155i)5-s + (0.949 − 0.314i)7-s + (0.188 − 0.391i)8-s + (−0.0200 − 0.0415i)10-s + (−0.0147 − 0.0117i)11-s + (0.850 + 0.678i)13-s + (1.20 + 0.446i)14-s + (1.10 − 0.530i)16-s + (−0.0969 + 0.424i)17-s + 1.25i·19-s + (0.00527 − 0.0231i)20-s + (−0.00541 − 0.0237i)22-s + (−0.864 + 0.197i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.701 - 0.713i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.701 - 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34063 + 0.981030i\)
\(L(\frac12)\) \(\approx\) \(2.34063 + 0.981030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 + 0.832i)T \)
good2 \( 1 + (-1.42 - 1.13i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (0.0720 + 0.0346i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (0.0489 + 0.0390i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-3.06 - 2.44i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.399 - 1.75i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 5.46iT - 19T^{2} \)
23 \( 1 + (4.14 - 0.945i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.80 + 2.01i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 6.38iT - 31T^{2} \)
37 \( 1 + (-0.385 + 1.68i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-6.08 - 2.92i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.79 + 0.866i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (2.99 - 3.76i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.0985 + 0.0224i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.52 - 1.69i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (3.20 + 0.731i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + (2.96 - 0.677i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.291 + 0.232i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (10.5 + 13.2i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-3.24 - 4.07i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 13.6iT - 97T^{2} \)
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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

−11.38779035526838962093200035267, −10.44980679313977470745845513969, −9.402790861160203784680455138858, −8.048349990866964940497210053132, −7.56251354977338490269763064846, −6.18759511352455439478720455488, −5.72844916599402127698826668594, −4.32398732212343332718749990099, −3.88451350270801563665287154630, −1.71959164625278906724499572415, 1.69513927521817392085022488308, 2.94929802367718366280401760321, 4.05878442352055617961457884663, 5.07975993591908140186501705231, 5.81467335901007124593484208236, 7.36979160728432185133582578169, 8.293268733390009096726118723436, 9.234118619794806525864086015315, 10.65274086532808393892374460476, 11.15895542935799063001793403267

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