Properties

Label 2-21e2-441.142-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.949 - 0.314i$
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  + (−0.489 + 1.24i)2-s + (−0.370 − 1.69i)3-s + (0.152 + 0.141i)4-s + (−1.59 + 0.770i)5-s + (2.28 + 0.366i)6-s + (1.91 + 1.82i)7-s + (−2.66 + 1.28i)8-s + (−2.72 + 1.25i)9-s + (−0.177 − 2.36i)10-s + (−1.93 − 2.43i)11-s + (0.182 − 0.310i)12-s + (−0.697 + 1.77i)13-s + (−3.21 + 1.48i)14-s + (1.89 + 2.42i)15-s + (−0.264 − 3.53i)16-s + (1.30 − 1.21i)17-s + ⋯
L(s)  = 1  + (−0.345 + 0.881i)2-s + (−0.213 − 0.976i)3-s + (0.0761 + 0.0706i)4-s + (−0.715 + 0.344i)5-s + (0.934 + 0.149i)6-s + (0.722 + 0.691i)7-s + (−0.941 + 0.453i)8-s + (−0.908 + 0.417i)9-s + (−0.0561 − 0.749i)10-s + (−0.584 − 0.732i)11-s + (0.0527 − 0.0895i)12-s + (−0.193 + 0.492i)13-s + (−0.859 + 0.397i)14-s + (0.489 + 0.625i)15-s + (−0.0661 − 0.882i)16-s + (0.317 − 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.949 - 0.314i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0922451 + 0.571129i\)
\(L(\frac12)\) \(\approx\) \(0.0922451 + 0.571129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.370 + 1.69i)T \)
7 \( 1 + (-1.91 - 1.82i)T \)
good2 \( 1 + (0.489 - 1.24i)T + (-1.46 - 1.36i)T^{2} \)
5 \( 1 + (1.59 - 0.770i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (1.93 + 2.43i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.697 - 1.77i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-1.30 + 1.21i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.00 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.48 - 6.51i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (8.28 + 2.55i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (1.17 - 2.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.32 + 1.02i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-9.54 - 6.51i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (5.37 - 3.66i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-2.35 + 6.00i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-5.96 + 1.83i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (4.76 - 3.24i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-5.52 + 5.12i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-6.50 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.35 - 10.3i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-2.62 - 0.395i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (4.61 + 7.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.832 - 2.12i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-5.06 - 12.9i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-8.43 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54775584845508720269541823920, −11.06977415399004081349699455882, −9.334288699080494223313696309288, −8.286273516218381592013948292428, −7.80661814774488720077936167345, −7.17091818216988836878760382015, −5.98506843696288917948957375636, −5.39287540551944005272664253362, −3.41929165617055022031564258492, −2.07114523412737426037328256196, 0.39134815136177240431318426777, 2.33548011984378882971589170813, 3.77631024391008764350475367202, 4.59453769444603812024473388766, 5.69116369570695226418503905947, 7.16015778149191304988665430831, 8.239544410878488515013274260920, 9.144004165737807570744421770172, 10.20461277434596934981340204456, 10.64262019650426380101754202758

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