Properties

Label 2-1110-111.68-c1-0-1
Degree $2$
Conductor $1110$
Sign $-0.364 - 0.931i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (−0.707 + 0.707i)2-s + (−1.73 − 0.0652i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (1.27 − 1.17i)6-s − 5.10·7-s + (0.707 + 0.707i)8-s + (2.99 + 0.225i)9-s + 1.00·10-s − 2.52·11-s + (−0.0652 + 1.73i)12-s + (4.82 − 4.82i)13-s + (3.60 − 3.60i)14-s + (1.17 + 1.27i)15-s − 1.00·16-s + (−2.16 − 2.16i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.999 − 0.0376i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.518 − 0.480i)6-s − 1.92·7-s + (0.250 + 0.250i)8-s + (0.997 + 0.0753i)9-s + 0.316·10-s − 0.760·11-s + (−0.0188 + 0.499i)12-s + (1.33 − 1.33i)13-s + (0.964 − 0.964i)14-s + (0.304 + 0.327i)15-s − 0.250·16-s + (−0.524 − 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.364 - 0.931i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1936065559\)
\(L(\frac12)\) \(\approx\) \(0.1936065559\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.73 + 0.0652i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (5.45 + 2.68i)T \)
good7 \( 1 + 5.10T + 7T^{2} \)
11 \( 1 + 2.52T + 11T^{2} \)
13 \( 1 + (-4.82 + 4.82i)T - 13iT^{2} \)
17 \( 1 + (2.16 + 2.16i)T + 17iT^{2} \)
19 \( 1 + (-2.84 + 2.84i)T - 19iT^{2} \)
23 \( 1 + (1.47 + 1.47i)T + 23iT^{2} \)
29 \( 1 + (4.98 - 4.98i)T - 29iT^{2} \)
31 \( 1 + (2.75 + 2.75i)T + 31iT^{2} \)
41 \( 1 - 4.87T + 41T^{2} \)
43 \( 1 + (6.39 - 6.39i)T - 43iT^{2} \)
47 \( 1 + 0.0691iT - 47T^{2} \)
53 \( 1 - 4.54iT - 53T^{2} \)
59 \( 1 + (-9.35 - 9.35i)T + 59iT^{2} \)
61 \( 1 + (-4.43 - 4.43i)T + 61iT^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 + 1.36iT - 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + (3.36 - 3.36i)T - 79iT^{2} \)
83 \( 1 + 9.73iT - 83T^{2} \)
89 \( 1 + (-7.24 + 7.24i)T - 89iT^{2} \)
97 \( 1 + (4.90 - 4.90i)T - 97iT^{2} \)
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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

−10.10163623730133085491950241550, −9.340495982939448594101679338984, −8.500985459262927812142010411323, −7.34654610706941109793069632772, −6.83835710302291501369702386106, −5.77375480952763844515808564285, −5.47745750999841054358125014710, −4.01950567730079168028670156660, −2.93231827494560732046147554720, −0.823338533532231652626009438234, 0.16569175597272743397496330329, 1.89678279107407582837871507426, 3.51440217897507360782129043719, 3.90488039975220823331700237043, 5.48446612092071690714779999006, 6.46313265895499544916857699044, 6.80815758561803678919202922431, 7.934711103525510626698749661248, 9.095942610032463159910990735138, 9.724730835082192825612316417098

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