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Properties

Label 2-1110-111.68-c1-0-22
Degree $2$
Conductor $1110$
Sign $0.968 + 0.249i$
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 + (−1.41 + i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.292 + 1.70i)6-s + 4·7-s + (−0.707 − 0.707i)8-s + (1.00 − 2.82i)9-s + 1.00·10-s + 4.24·11-s + (1.00 + 1.41i)12-s + (−1 + i)13-s + (2.82 − 2.82i)14-s + (−1.70 − 0.292i)15-s − 1.00·16-s + (−1.41 − 1.41i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.816 + 0.577i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.119 + 0.696i)6-s + 1.51·7-s + (−0.250 − 0.250i)8-s + (0.333 − 0.942i)9-s + 0.316·10-s + 1.27·11-s + (0.288 + 0.408i)12-s + (−0.277 + 0.277i)13-s + (0.755 − 0.755i)14-s + (−0.440 − 0.0756i)15-s − 0.250·16-s + (−0.342 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.968 + 0.249i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.968 + 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.116769644\)
\(L(\frac12)\) \(\approx\) \(2.116769644\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-6 - i)T \)
good7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 + (2 - 2i)T - 19iT^{2} \)
23 \( 1 + (4.24 + 4.24i)T + 23iT^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 + (-9 + 9i)T - 43iT^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 + 2.82iT - 53T^{2} \)
59 \( 1 + (2.82 + 2.82i)T + 59iT^{2} \)
61 \( 1 + (-4 - 4i)T + 61iT^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + (3 - 3i)T - 79iT^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + (12.7 - 12.7i)T - 89iT^{2} \)
97 \( 1 + (12 - 12i)T - 97iT^{2} \)
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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

−10.05005624941594041167221492047, −9.219680159731945669645644213527, −8.330614580725674195892196500729, −7.00689498302586729877481128948, −6.24572512666660200925440454367, −5.38991336622655128978607798956, −4.42963503383769462053577511621, −4.01918135065101729790748411243, −2.36074257945895288819541044737, −1.18344050296027942264272713827, 1.20503683304256837796428141553, 2.27528615705376185896537937589, 4.27268727721880451152453244412, 4.65415851961795717985334539737, 5.83146757393035621026640370164, 6.22155358515797576018727417546, 7.41769432454667383873655882936, 7.930303445439362989729878090617, 8.845001705555601930792803767265, 9.870729688487137166473627115870

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