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Properties

Label 2-21e2-49.2-c1-0-9
Degree $2$
Conductor $441$
Sign $0.330 - 0.943i$
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  + (2.50 + 0.772i)2-s + (4.02 + 2.74i)4-s + (−2.04 + 0.307i)5-s + (0.189 + 2.63i)7-s + (4.69 + 5.89i)8-s + (−5.35 − 0.806i)10-s + (1.36 − 1.27i)11-s + (−0.677 − 2.96i)13-s + (−1.56 + 6.75i)14-s + (3.65 + 9.32i)16-s + (0.104 + 1.40i)17-s + (2.73 − 4.73i)19-s + (−9.06 − 4.36i)20-s + (4.41 − 2.12i)22-s + (0.0152 − 0.203i)23-s + ⋯
L(s)  = 1  + (1.77 + 0.546i)2-s + (2.01 + 1.37i)4-s + (−0.913 + 0.137i)5-s + (0.0717 + 0.997i)7-s + (1.66 + 2.08i)8-s + (−1.69 − 0.255i)10-s + (0.413 − 0.383i)11-s + (−0.187 − 0.822i)13-s + (−0.417 + 1.80i)14-s + (0.914 + 2.33i)16-s + (0.0254 + 0.339i)17-s + (0.626 − 1.08i)19-s + (−2.02 − 0.976i)20-s + (0.941 − 0.453i)22-s + (0.00317 − 0.0423i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74562 + 1.94675i\)
\(L(\frac12)\) \(\approx\) \(2.74562 + 1.94675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.189 - 2.63i)T \)
good2 \( 1 + (-2.50 - 0.772i)T + (1.65 + 1.12i)T^{2} \)
5 \( 1 + (2.04 - 0.307i)T + (4.77 - 1.47i)T^{2} \)
11 \( 1 + (-1.36 + 1.27i)T + (0.822 - 10.9i)T^{2} \)
13 \( 1 + (0.677 + 2.96i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.104 - 1.40i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (-2.73 + 4.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0152 + 0.203i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (1.62 + 0.781i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (2.57 + 4.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.23 + 4.25i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (-5.90 - 7.40i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-1.59 + 2.00i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (1.58 + 0.490i)T + (38.8 + 26.4i)T^{2} \)
53 \( 1 + (5.98 + 4.07i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (8.25 + 1.24i)T + (56.3 + 17.3i)T^{2} \)
61 \( 1 + (5.45 - 3.72i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (1.78 + 3.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.02 + 2.89i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.168 - 0.0521i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (7.82 - 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.14 - 9.39i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.911 + 0.845i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−11.40572502546314889561358282629, −11.20259040297478338184972564087, −9.371751443097187982004458204234, −8.071174315715901766254538517894, −7.46769942035291504532614905757, −6.28632183218208621680987558898, −5.55577512620201715195396652853, −4.55412148470207048086733912589, −3.51641911714179073234242165098, −2.59688256144208140089805925527, 1.56676261853200040727686715440, 3.25838163918685891180013083563, 4.11871840626199070107358819892, 4.68916642555760369439087850823, 5.99298078904917285573827378187, 7.05587098637996017980592435015, 7.73431340319390719556412698626, 9.466447210050444220813375519792, 10.50806432819830624923476397211, 11.31401906467288448618627285717

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