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Properties

Label 2-1110-37.27-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.367 - 0.929i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 0.999i·6-s + (−0.366 − 0.633i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + 0.267·11-s + (−0.499 + 0.866i)12-s + (1.5 − 0.866i)13-s − 0.732i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (6.23 + 3.59i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + 0.408i·6-s + (−0.138 − 0.239i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s + 0.0807·11-s + (−0.144 + 0.249i)12-s + (0.416 − 0.240i)13-s − 0.195i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (1.51 + 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.367 - 0.929i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.804975592\)
\(L(\frac12)\) \(\approx\) \(2.804975592\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.5 - 6.06i)T \)
good7 \( 1 + (0.366 + 0.633i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.267T + 11T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.23 - 3.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 4.26iT - 29T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
41 \( 1 + (4.09 + 7.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.535iT - 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 + (-4.09 + 7.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.401 + 0.232i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.09 + 0.633i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.33 + 9.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 + 7.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (4.26 - 2.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.09 + 5.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.1 - 6.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−10.16647943178021192101305782708, −9.071025374625231839323853737096, −8.367346969251830059032074320485, −7.49280866734735931244452070384, −6.53705088469656180980822695143, −5.55434903294249812600614750907, −4.97893996519387419913420726686, −3.71654032140345851040868495266, −3.16099960036251951844973149412, −1.54431912385779591712785244156, 1.13955285198319872459517780076, 2.42236356007080368059032455355, 3.23572598011860907327477413656, 4.33395475643929043177280164504, 5.60240420585387882821934734226, 6.08385233854807579923155978356, 7.17146187175655986042039315054, 7.87400990766636687355188645462, 9.025339875091621655616948331143, 9.747164816164051138800492628157

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