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Properties

Label 2-21e2-7.4-c3-0-43
Degree $2$
Conductor $441$
Sign $-0.991 + 0.126i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)4-s − 70·13-s + (−31.9 − 55.4i)16-s + (−28 − 48.4i)19-s + (62.5 − 108. i)25-s + (−154 + 266. i)31-s + (−55 − 95.2i)37-s − 520·43-s + (−280 + 484. i)52-s + (−91 − 157. i)61-s − 511.·64-s + (440 − 762. i)67-s + (−595 + 1.03e3i)73-s − 448·76-s + (−442 − 765. i)79-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s − 1.49·13-s + (−0.499 − 0.866i)16-s + (−0.338 − 0.585i)19-s + (0.5 − 0.866i)25-s + (−0.892 + 1.54i)31-s + (−0.244 − 0.423i)37-s − 1.84·43-s + (−0.746 + 1.29i)52-s + (−0.191 − 0.330i)61-s − 0.999·64-s + (0.802 − 1.38i)67-s + (−0.953 + 1.65i)73-s − 0.676·76-s + (−0.629 − 1.09i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7386172822\)
\(L(\frac12)\) \(\approx\) \(0.7386172822\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 70T + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (28 + 48.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (55 + 95.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 520T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (91 + 157. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-440 + 762. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + (595 - 1.03e3i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (442 + 765. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−10.27956083209803175970728823408, −9.567272421880884948901266038398, −8.529049672316125847966618328334, −7.22313989641582163104713892595, −6.64668630759271982649604512118, −5.39489699364415509355371136752, −4.65829605843546070449210196188, −2.92776921553246232766972661783, −1.80493315745799743006436770142, −0.21407224702063321874449893990, 1.94842047405757395210193268107, 3.05964471367929791532014632990, 4.20949255296742531605535240181, 5.41828436421701897590208947581, 6.70967973899054809091711450815, 7.45779498421284055779773259075, 8.251388036442077242208349159031, 9.332757193244271912365143230436, 10.25081169568286480137101051612, 11.30453194257185347936891286881

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