Properties

Label 2-21e2-63.4-c1-0-35
Degree $2$
Conductor $441$
Sign $-0.0632 - 0.997i$
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  + (1.35 − 2.35i)2-s + (−1.69 − 0.374i)3-s + (−2.68 − 4.65i)4-s − 1.58·5-s + (−3.17 + 3.46i)6-s − 9.15·8-s + (2.72 + 1.26i)9-s + (−2.15 + 3.73i)10-s − 1.34·11-s + (2.80 + 8.87i)12-s + (1.58 − 2.75i)13-s + (2.68 + 0.593i)15-s + (−7.05 + 12.2i)16-s + (−1.40 + 2.42i)17-s + (6.66 − 4.67i)18-s + (−0.312 − 0.541i)19-s + ⋯
L(s)  = 1  + (0.959 − 1.66i)2-s + (−0.976 − 0.215i)3-s + (−1.34 − 2.32i)4-s − 0.709·5-s + (−1.29 + 1.41i)6-s − 3.23·8-s + (0.906 + 0.421i)9-s + (−0.681 + 1.17i)10-s − 0.406·11-s + (0.808 + 2.56i)12-s + (0.440 − 0.763i)13-s + (0.692 + 0.153i)15-s + (−1.76 + 3.05i)16-s + (−0.339 + 0.588i)17-s + (1.57 − 1.10i)18-s + (−0.0717 − 0.124i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0632 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0632 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0632 - 0.997i$
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: \((2,\ 441,\ (\ :1/2),\ -0.0632 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493575 + 0.525866i\)
\(L(\frac12)\) \(\approx\) \(0.493575 + 0.525866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.374i)T \)
7 \( 1 \)
good2 \( 1 + (-1.35 + 2.35i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
13 \( 1 + (-1.58 + 2.75i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.40 - 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.312 + 0.541i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.284T + 23T^{2} \)
29 \( 1 + (-2.27 - 3.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.71 + 6.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.01 + 6.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.01 + 8.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.12 + 5.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.57 - 9.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.39 - 2.41i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.28 + 3.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.192 + 0.333i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.26 - 2.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.45T + 71T^{2} \)
73 \( 1 + (-0.234 + 0.405i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.85 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.99 + 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.29 - 2.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.22 - 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.80200337881744785891898952259, −10.22496566554022741942898094594, −9.016661394050982794457623232561, −7.63415907208498789991201753120, −6.13577426169408301070287796322, −5.37578637189079845317700075428, −4.35889541605305098919093245774, −3.48765716394718261214297779935, −1.95234251528966147221861941354, −0.37384344523456764038709977045, 3.49137963851531131061068674527, 4.46665898402681637378011673382, 5.12978843297383545134814553155, 6.24253254365716432667064934366, 6.85925630305912958357780617775, 7.77226476933783039599379387087, 8.667670042629493393585488255191, 9.842305609883955604529644360440, 11.35003940971560474146001953358, 11.88488150233118772200709964590

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