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Properties

Label 2-21e2-63.4-c1-0-32
Degree $2$
Conductor $441$
Sign $-0.996 + 0.0871i$
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  + (1.19 − 2.06i)2-s + (−0.266 − 1.71i)3-s + (−1.84 − 3.20i)4-s + 2.92·5-s + (−3.85 − 1.49i)6-s − 4.05·8-s + (−2.85 + 0.913i)9-s + (3.48 − 6.03i)10-s − 1.35·11-s + (−4.98 + 4.01i)12-s + (0.733 − 1.26i)13-s + (−0.779 − 4.99i)15-s + (−1.13 + 1.96i)16-s + (−1.65 + 2.86i)17-s + (−1.52 + 6.99i)18-s + (1.10 + 1.91i)19-s + ⋯
L(s)  = 1  + (0.843 − 1.46i)2-s + (−0.154 − 0.988i)3-s + (−0.924 − 1.60i)4-s + 1.30·5-s + (−1.57 − 0.608i)6-s − 1.43·8-s + (−0.952 + 0.304i)9-s + (1.10 − 1.90i)10-s − 0.408·11-s + (−1.43 + 1.16i)12-s + (0.203 − 0.352i)13-s + (−0.201 − 1.29i)15-s + (−0.284 + 0.492i)16-s + (−0.401 + 0.695i)17-s + (−0.358 + 1.64i)18-s + (0.253 + 0.438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.996 + 0.0871i$
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: \((2,\ 441,\ (\ :1/2),\ -0.996 + 0.0871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0990757 - 2.27036i\)
\(L(\frac12)\) \(\approx\) \(0.0990757 - 2.27036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.266 + 1.71i)T \)
7 \( 1 \)
good2 \( 1 + (-1.19 + 2.06i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.92T + 5T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
13 \( 1 + (-0.733 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.65 - 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 + (-0.521 - 0.903i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.63 - 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.904 + 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.98 + 3.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.22 - 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.10 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.279 + 0.484i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.40 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (5.22 - 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.383 - 0.664i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.983 - 1.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.20 + 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.14 - 7.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.80064240307821876552697325088, −10.20084561919401009298678921283, −9.200891820077509658778754642659, −8.020063327002721028665534361863, −6.54147532633248592723524903848, −5.72344787854010015422994409301, −4.89258361726246329579322976640, −3.22936389945153075996082688145, −2.23878908587145084296389749933, −1.29946294988008338269849896245, 2.75197051390792007922040285756, 4.22713766381551408712530309213, 5.07485166248432895662537006602, 5.80897445131117098735427021722, 6.52407512397442680825400851735, 7.67555546310905823429006590910, 8.924121038059772612386207971066, 9.484096465425257286235558584185, 10.54387350164583462131382089677, 11.54942527881044724645856457986

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