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Properties

Label 2-21e2-147.104-c1-0-15
Degree $2$
Conductor $441$
Sign $-0.657 + 0.753i$
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  + (−0.00441 − 0.00351i)2-s + (−0.445 − 1.94i)4-s + (−1.95 − 0.943i)5-s + (2.55 + 0.694i)7-s + (−0.00979 + 0.0203i)8-s + (0.00532 + 0.0110i)10-s + (−1.56 − 1.24i)11-s + (−1.10 − 0.883i)13-s + (−0.00881 − 0.0120i)14-s + (−3.60 + 1.73i)16-s + (0.751 − 3.29i)17-s − 6.30i·19-s + (−0.967 + 4.23i)20-s + (0.00250 + 0.0109i)22-s + (−7.96 + 1.81i)23-s + ⋯
L(s)  = 1  + (−0.00311 − 0.00248i)2-s + (−0.222 − 0.974i)4-s + (−0.876 − 0.421i)5-s + (0.964 + 0.262i)7-s + (−0.00346 + 0.00718i)8-s + (0.00168 + 0.00349i)10-s + (−0.470 − 0.375i)11-s + (−0.307 − 0.245i)13-s + (−0.00235 − 0.00321i)14-s + (−0.900 + 0.433i)16-s + (0.182 − 0.798i)17-s − 1.44i·19-s + (−0.216 + 0.948i)20-s + (0.000534 + 0.00234i)22-s + (−1.66 + 0.378i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.657 + 0.753i$
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: \((2,\ 441,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363217 - 0.798927i\)
\(L(\frac12)\) \(\approx\) \(0.363217 - 0.798927i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 - 0.694i)T \)
good2 \( 1 + (0.00441 + 0.00351i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (1.95 + 0.943i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.56 + 1.24i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.10 + 0.883i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.751 + 3.29i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.30iT - 19T^{2} \)
23 \( 1 + (7.96 - 1.81i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (5.47 + 1.24i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 5.21iT - 31T^{2} \)
37 \( 1 + (-1.55 + 6.79i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-9.97 - 4.80i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.0684 + 0.0329i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.46 + 4.34i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.739 + 0.168i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-9.30 + 4.47i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-7.51 - 1.71i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 + (-14.3 + 3.26i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.21 + 1.76i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + (-1.21 - 1.52i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.85 + 6.09i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 0.0843iT - 97T^{2} \)
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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

−11.01372967628918213609085166413, −9.867457119270061724991118392791, −8.971906736127380965350595075948, −8.088349839143552306520850065063, −7.25318288607080814289982850073, −5.76751246736616212517151065710, −5.03089553396900004114275940450, −4.12711091152345775401758497499, −2.29371289256281097437588011382, −0.54909026147025563400051465151, 2.18416612459875140622850922189, 3.80910912775100539345266965757, 4.25131614691439138543823687556, 5.73861070959363746620957891924, 7.22880459151220996826407653282, 7.907764757740664934762500752715, 8.267494192397793606028546027527, 9.693762355739001098803819740144, 10.69773253141037483892014348541, 11.58636954713237666767209662310

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