Properties

Label 2-1110-111.68-c1-0-32
Degree $2$
Conductor $1110$
Sign $0.545 + 0.838i$
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 + (1.33 − 1.10i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.159 + 1.72i)6-s − 3.17·7-s + (0.707 + 0.707i)8-s + (0.549 − 2.94i)9-s − 1.00·10-s + 4.29·11-s + (−1.10 − 1.33i)12-s + (0.469 − 0.469i)13-s + (2.24 − 2.24i)14-s + (1.72 + 0.159i)15-s − 1.00·16-s + (−4.17 − 4.17i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.769 − 0.639i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.0650 + 0.704i)6-s − 1.19·7-s + (0.250 + 0.250i)8-s + (0.183 − 0.983i)9-s − 0.316·10-s + 1.29·11-s + (−0.319 − 0.384i)12-s + (0.130 − 0.130i)13-s + (0.599 − 0.599i)14-s + (0.445 + 0.0411i)15-s − 0.250·16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.545 + 0.838i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.464231431\)
\(L(\frac12)\) \(\approx\) \(1.464231431\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 1.10i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-3.36 - 5.06i)T \)
good7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 + (-0.469 + 0.469i)T - 13iT^{2} \)
17 \( 1 + (4.17 + 4.17i)T + 17iT^{2} \)
19 \( 1 + (-2.35 + 2.35i)T - 19iT^{2} \)
23 \( 1 + (2.43 + 2.43i)T + 23iT^{2} \)
29 \( 1 + (-3.28 + 3.28i)T - 29iT^{2} \)
31 \( 1 + (0.0756 + 0.0756i)T + 31iT^{2} \)
41 \( 1 - 6.28T + 41T^{2} \)
43 \( 1 + (-2.94 + 2.94i)T - 43iT^{2} \)
47 \( 1 + 0.554iT - 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 + (-5.42 - 5.42i)T + 59iT^{2} \)
61 \( 1 + (3.80 + 3.80i)T + 61iT^{2} \)
67 \( 1 + 7.22iT - 67T^{2} \)
71 \( 1 + 8.00iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + (2.36 - 2.36i)T - 79iT^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-2.62 + 2.62i)T - 89iT^{2} \)
97 \( 1 + (0.130 - 0.130i)T - 97iT^{2} \)
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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

−9.417198914422842449258506044020, −9.055932332526140668259775192721, −8.084450652629747755683695680662, −6.98908418020245109875519014947, −6.66587738978822742784921957266, −5.96629990930810180734537828847, −4.38045162554329288022718950799, −3.21756063915729940801057934245, −2.26581832549843367568132288144, −0.72849740035425698427832363658, 1.48523313045275244095535275236, 2.69537978192420827742575129163, 3.75245802732918922893287483342, 4.27581671695999378085161524626, 5.82742982742705327040269337909, 6.67692873139692238944924809620, 7.74725712058068532792734119888, 8.758429717361086150180640519467, 9.223255513274989797546179467183, 9.767026877367524326121157423513

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