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Properties

Label 2-1110-185.43-c1-0-14
Degree $2$
Conductor $1110$
Sign $0.338 - 0.941i$
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  + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−2.19 + 0.407i)5-s + (−0.707 − 0.707i)6-s + (1.64 − 1.64i)7-s − i·8-s − 1.00i·9-s + (−0.407 − 2.19i)10-s + 5.09i·11-s + (0.707 − 0.707i)12-s − 5.90i·13-s + (1.64 + 1.64i)14-s + (1.26 − 1.84i)15-s + 16-s + 0.720·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.983 + 0.182i)5-s + (−0.288 − 0.288i)6-s + (0.621 − 0.621i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.128 − 0.695i)10-s + 1.53i·11-s + (0.204 − 0.204i)12-s − 1.63i·13-s + (0.439 + 0.439i)14-s + (0.327 − 0.475i)15-s + 0.250·16-s + 0.174·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.338 - 0.941i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.338 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.140770011\)
\(L(\frac12)\) \(\approx\) \(1.140770011\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.19 - 0.407i)T \)
37 \( 1 + (-6.04 + 0.654i)T \)
good7 \( 1 + (-1.64 + 1.64i)T - 7iT^{2} \)
11 \( 1 - 5.09iT - 11T^{2} \)
13 \( 1 + 5.90iT - 13T^{2} \)
17 \( 1 - 0.720T + 17T^{2} \)
19 \( 1 + (-5.20 + 5.20i)T - 19iT^{2} \)
23 \( 1 - 2.47iT - 23T^{2} \)
29 \( 1 + (-2.09 - 2.09i)T + 29iT^{2} \)
31 \( 1 + (6.99 - 6.99i)T - 31iT^{2} \)
41 \( 1 + 6.58iT - 41T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 + (-6.69 + 6.69i)T - 47iT^{2} \)
53 \( 1 + (-6.02 - 6.02i)T + 53iT^{2} \)
59 \( 1 + (-1.31 + 1.31i)T - 59iT^{2} \)
61 \( 1 + (0.136 - 0.136i)T - 61iT^{2} \)
67 \( 1 + (-8.82 - 8.82i)T + 67iT^{2} \)
71 \( 1 - 0.630T + 71T^{2} \)
73 \( 1 + (0.800 - 0.800i)T - 73iT^{2} \)
79 \( 1 + (1.35 - 1.35i)T - 79iT^{2} \)
83 \( 1 + (-0.0877 - 0.0877i)T + 83iT^{2} \)
89 \( 1 + (0.651 + 0.651i)T + 89iT^{2} \)
97 \( 1 + 5.99T + 97T^{2} \)
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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

−10.06133007771189880588460562385, −9.139911682670902000671767626698, −8.104670184583189249531126765249, −7.25977440430436385160831937206, −7.14373136269497788831922744907, −5.52348478310530305086754219582, −4.91055404573278215481306800307, −4.12120771184491357683220967595, −3.05541179276204492861213695890, −0.882598759820739996538064417240, 0.821385669746681824531597731291, 2.10316294003457326203358454050, 3.45621403519220047305734730478, 4.28412304594394073141042947939, 5.36404037606424219872042940968, 6.14045203675691910334728545810, 7.38531428116214734701113395548, 8.176198443271464357709454326555, 8.782930004452520568117088835294, 9.657042418767450827398974716516

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