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Properties

Label 2-1110-185.184-c1-0-20
Degree $2$
Conductor $1110$
Sign $0.363 + 0.931i$
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  − 2-s − i·3-s + 4-s + (−0.0603 + 2.23i)5-s + i·6-s − 3.18i·7-s − 8-s − 9-s + (0.0603 − 2.23i)10-s + 2.65·11-s − i·12-s − 0.269·13-s + 3.18i·14-s + (2.23 + 0.0603i)15-s + 16-s + 1.00·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (−0.0269 + 0.999i)5-s + 0.408i·6-s − 1.20i·7-s − 0.353·8-s − 0.333·9-s + (0.0190 − 0.706i)10-s + 0.800·11-s − 0.288i·12-s − 0.0747·13-s + 0.850i·14-s + (0.577 + 0.0155i)15-s + 0.250·16-s + 0.244·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.097540075\)
\(L(\frac12)\) \(\approx\) \(1.097540075\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + iT \)
5 \( 1 + (0.0603 - 2.23i)T \)
37 \( 1 + (2.36 + 5.60i)T \)
good7 \( 1 + 3.18iT - 7T^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + 0.269T + 13T^{2} \)
17 \( 1 - 1.00T + 17T^{2} \)
19 \( 1 + 0.921iT - 19T^{2} \)
23 \( 1 - 2.37T + 23T^{2} \)
29 \( 1 + 1.10iT - 29T^{2} \)
31 \( 1 + 3.57iT - 31T^{2} \)
41 \( 1 - 4.25T + 41T^{2} \)
43 \( 1 - 6.87T + 43T^{2} \)
47 \( 1 + 3.07iT - 47T^{2} \)
53 \( 1 + 3.55iT - 53T^{2} \)
59 \( 1 - 3.24iT - 59T^{2} \)
61 \( 1 + 2.85iT - 61T^{2} \)
67 \( 1 - 4.14iT - 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 2.97iT - 73T^{2} \)
79 \( 1 + 10.1iT - 79T^{2} \)
83 \( 1 + 9.43iT - 83T^{2} \)
89 \( 1 + 12.2iT - 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−9.741188993128402299417801902412, −8.896934807841848738577139204758, −7.76140286633866839846987436154, −7.27550692667499370590562513695, −6.66236198013130685681557733855, −5.81267369319240986164011176999, −4.20070544727581339125612954579, −3.26020687147729042450723656075, −2.04478303693920567012788004888, −0.69948545797134851628222418044, 1.23551284859491703003059438118, 2.55611846416212378552712866552, 3.81641118129563654747045580555, 4.96515842677623434219065886848, 5.68851920008828498388946610737, 6.61049675424878877005119989248, 7.87565307584513068042424517956, 8.614177967074610361766984787676, 9.200027936252811149959581882803, 9.633330022270044607903856515535

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