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Properties

Label 2-1110-111.68-c1-0-33
Degree $2$
Conductor $1110$
Sign $-0.141 + 0.989i$
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 + (−0.407 − 1.68i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (1.47 + 0.902i)6-s + 2.94·7-s + (0.707 + 0.707i)8-s + (−2.66 + 1.37i)9-s − 1.00·10-s − 5.04·11-s + (−1.68 + 0.407i)12-s + (2.73 − 2.73i)13-s + (−2.07 + 2.07i)14-s + (0.902 − 1.47i)15-s − 1.00·16-s + (−5.09 − 5.09i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.235 − 0.971i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.603 + 0.368i)6-s + 1.11·7-s + (0.250 + 0.250i)8-s + (−0.889 + 0.457i)9-s − 0.316·10-s − 1.52·11-s + (−0.485 + 0.117i)12-s + (0.758 − 0.758i)13-s + (−0.555 + 0.555i)14-s + (0.232 − 0.381i)15-s − 0.250·16-s + (−1.23 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.141 + 0.989i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9570060238\)
\(L(\frac12)\) \(\approx\) \(0.9570060238\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.407 + 1.68i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (5.76 - 1.93i)T \)
good7 \( 1 - 2.94T + 7T^{2} \)
11 \( 1 + 5.04T + 11T^{2} \)
13 \( 1 + (-2.73 + 2.73i)T - 13iT^{2} \)
17 \( 1 + (5.09 + 5.09i)T + 17iT^{2} \)
19 \( 1 + (-2.15 + 2.15i)T - 19iT^{2} \)
23 \( 1 + (-3.53 - 3.53i)T + 23iT^{2} \)
29 \( 1 + (-4.90 + 4.90i)T - 29iT^{2} \)
31 \( 1 + (4.78 + 4.78i)T + 31iT^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 + (1.16 - 1.16i)T - 43iT^{2} \)
47 \( 1 + 6.36iT - 47T^{2} \)
53 \( 1 + 3.67iT - 53T^{2} \)
59 \( 1 + (6.34 + 6.34i)T + 59iT^{2} \)
61 \( 1 + (-3.36 - 3.36i)T + 61iT^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 6.94iT - 71T^{2} \)
73 \( 1 - 8.41iT - 73T^{2} \)
79 \( 1 + (-10.3 + 10.3i)T - 79iT^{2} \)
83 \( 1 + 4.78iT - 83T^{2} \)
89 \( 1 + (2.54 - 2.54i)T - 89iT^{2} \)
97 \( 1 + (9.97 - 9.97i)T - 97iT^{2} \)
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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

−9.456226615774883134452186961570, −8.516468568719562115902174118635, −7.84094011275792086229413518510, −7.31457649263899105679753987182, −6.40239461560434064809518855402, −5.36657999875725484395840992267, −4.96344015230763602626636402226, −2.88860861296958081285488076826, −1.95625283149115142992809060645, −0.51793879155400736481907719605, 1.49930670795466634652165909409, 2.71322388196575191870196163370, 4.00157368500669505420119367769, 4.80847850142146220148255871415, 5.54711827640432961392894597640, 6.72282531953146038166986067367, 8.030707250146900146369872592150, 8.645530968693853814021987035323, 9.137824048446037760660983769792, 10.36737096631391991166154335193

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