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Properties

Label 2-1110-185.142-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.0183 - 0.999i$
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 + (1.71 − 1.43i)5-s + (0.707 − 0.707i)6-s + (3.31 + 3.31i)7-s − i·8-s + 1.00i·9-s + (1.43 + 1.71i)10-s + 4.55i·11-s + (0.707 + 0.707i)12-s + 0.0409i·13-s + (−3.31 + 3.31i)14-s + (−2.22 − 0.199i)15-s + 16-s − 3.39·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.767 − 0.641i)5-s + (0.288 − 0.288i)6-s + (1.25 + 1.25i)7-s − 0.353i·8-s + 0.333i·9-s + (0.453 + 0.542i)10-s + 1.37i·11-s + (0.204 + 0.204i)12-s + 0.0113i·13-s + (−0.885 + 0.885i)14-s + (−0.575 − 0.0514i)15-s + 0.250·16-s − 0.822·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0183 - 0.999i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.0183 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.617874307\)
\(L(\frac12)\) \(\approx\) \(1.617874307\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 1.43i)T \)
37 \( 1 + (1.32 + 5.93i)T \)
good7 \( 1 + (-3.31 - 3.31i)T + 7iT^{2} \)
11 \( 1 - 4.55iT - 11T^{2} \)
13 \( 1 - 0.0409iT - 13T^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
19 \( 1 + (1.42 + 1.42i)T + 19iT^{2} \)
23 \( 1 - 6.74iT - 23T^{2} \)
29 \( 1 + (-2.73 + 2.73i)T - 29iT^{2} \)
31 \( 1 + (-2.17 - 2.17i)T + 31iT^{2} \)
41 \( 1 - 4.17iT - 41T^{2} \)
43 \( 1 - 8.29iT - 43T^{2} \)
47 \( 1 + (-1.79 - 1.79i)T + 47iT^{2} \)
53 \( 1 + (5.35 - 5.35i)T - 53iT^{2} \)
59 \( 1 + (7.59 + 7.59i)T + 59iT^{2} \)
61 \( 1 + (-4.71 - 4.71i)T + 61iT^{2} \)
67 \( 1 + (-10.3 + 10.3i)T - 67iT^{2} \)
71 \( 1 - 3.26T + 71T^{2} \)
73 \( 1 + (-3.99 - 3.99i)T + 73iT^{2} \)
79 \( 1 + (3.34 + 3.34i)T + 79iT^{2} \)
83 \( 1 + (-1.21 + 1.21i)T - 83iT^{2} \)
89 \( 1 + (9.81 - 9.81i)T - 89iT^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−9.691454870674575582182528314180, −9.172490949833814591903100045080, −8.304936068687294024075238802658, −7.63254731640293747795087698803, −6.56330287248879324104013070039, −5.80013875134754393383754688497, −4.95303217069590826367471600284, −4.57050488201185187138353711157, −2.34015244578910255028165043151, −1.54315623714510599465816372360, 0.801970490310645066243087810967, 2.12116018985054768111219748631, 3.39314326558217559472396905810, 4.33907619254214670946764853685, 5.15343077207722144359781034223, 6.20071722497390909130528649473, 7.02828030195235088813515521029, 8.271537496088439552318760407243, 8.822827822449334062601083264016, 10.15308696647593337337604698057

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