Properties

Label 2-21e2-63.4-c1-0-1
Degree $2$
Conductor $441$
Sign $-0.820 + 0.571i$
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  + (−0.0341 + 0.0592i)2-s + (−1.15 + 1.29i)3-s + (0.997 + 1.72i)4-s − 2.66·5-s + (−0.0368 − 0.112i)6-s − 0.273·8-s + (−0.329 − 2.98i)9-s + (0.0910 − 0.157i)10-s − 1.59·11-s + (−3.38 − 0.709i)12-s + (−2.62 + 4.54i)13-s + (3.07 − 3.43i)15-s + (−1.98 + 3.43i)16-s + (3.27 − 5.67i)17-s + (0.187 + 0.0824i)18-s + (−0.950 − 1.64i)19-s + ⋯
L(s)  = 1  + (−0.0241 + 0.0418i)2-s + (−0.667 + 0.744i)3-s + (0.498 + 0.864i)4-s − 1.19·5-s + (−0.0150 − 0.0459i)6-s − 0.0965·8-s + (−0.109 − 0.993i)9-s + (0.0287 − 0.0498i)10-s − 0.482·11-s + (−0.976 − 0.204i)12-s + (−0.728 + 1.26i)13-s + (0.794 − 0.887i)15-s + (−0.496 + 0.859i)16-s + (0.793 − 1.37i)17-s + (0.0442 + 0.0194i)18-s + (−0.218 − 0.377i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.820 + 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0786142 - 0.250269i\)
\(L(\frac12)\) \(\approx\) \(0.0786142 - 0.250269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 - 1.29i)T \)
7 \( 1 \)
good2 \( 1 + (0.0341 - 0.0592i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.66T + 5T^{2} \)
11 \( 1 + 1.59T + 11T^{2} \)
13 \( 1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.27 + 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.950 + 1.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.06T + 23T^{2} \)
29 \( 1 + (3.19 + 5.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.35 + 5.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.69 - 6.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.89 - 3.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.44 - 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.44 - 9.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.09 - 1.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.406 - 0.704i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.41 + 5.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.235 + 0.407i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.65773499818108118115395539099, −11.14863950344585558986075909209, −9.855970324062256763655084501576, −9.034146536378036668989905426378, −7.72976839180350129957134398613, −7.28845319176662271387600906272, −6.05820321049261821225996271836, −4.63024953227338370651395872032, −3.97534928549390843888308378425, −2.75066725306946431375117667409, 0.17084859042079694861917161173, 1.81420424477388257931260640107, 3.45515916790747738433068196661, 5.12623086149280330557653863970, 5.73974949286270062210375115917, 6.94393676661131074972063152127, 7.67765753481683185184999548834, 8.426446250196352621879262448245, 10.26252034947111585047396445891, 10.52806223787066819209099927869

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