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Properties

Label 2-1110-5.4-c1-0-17
Degree $2$
Conductor $1110$
Sign $0.193 + 0.981i$
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 − i·3-s − 4-s + (−2.19 + 0.432i)5-s − 6-s + 1.86i·7-s + i·8-s − 9-s + (0.432 + 2.19i)10-s + 3.38·11-s + i·12-s + 2.76i·13-s + 1.86·14-s + (0.432 + 2.19i)15-s + 16-s − 7.38i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.981 + 0.193i)5-s − 0.408·6-s + 0.704i·7-s + 0.353i·8-s − 0.333·9-s + (0.136 + 0.693i)10-s + 1.02·11-s + 0.288i·12-s + 0.765i·13-s + 0.498·14-s + (0.111 + 0.566i)15-s + 0.250·16-s − 1.79i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.193 + 0.981i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267708072\)
\(L(\frac12)\) \(\approx\) \(1.267708072\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (2.19 - 0.432i)T \)
37 \( 1 + iT \)
good7 \( 1 - 1.86iT - 7T^{2} \)
11 \( 1 - 3.38T + 11T^{2} \)
13 \( 1 - 2.76iT - 13T^{2} \)
17 \( 1 + 7.38iT - 17T^{2} \)
19 \( 1 + 0.373T + 19T^{2} \)
23 \( 1 - 2.10iT - 23T^{2} \)
29 \( 1 - 9.49T + 29T^{2} \)
31 \( 1 - 0.896T + 31T^{2} \)
41 \( 1 - 3.28T + 41T^{2} \)
43 \( 1 + 4.42iT - 43T^{2} \)
47 \( 1 + 6.38iT - 47T^{2} \)
53 \( 1 + 0.523iT - 53T^{2} \)
59 \( 1 - 8.11T + 59T^{2} \)
61 \( 1 + 7.87T + 61T^{2} \)
67 \( 1 + 13.5iT - 67T^{2} \)
71 \( 1 + 0.929T + 71T^{2} \)
73 \( 1 + 0.103iT - 73T^{2} \)
79 \( 1 - 4.86T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 - 4.28T + 89T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−9.484025242111639700786165912468, −8.957005153071706921243327841361, −8.136330264948721944470318262921, −7.13299727343037784197714893308, −6.50489521844585848152489045110, −5.18629638742659682837308997974, −4.27320964640238978201622009263, −3.22181982398148663184290303856, −2.27044828999338320780697383616, −0.800102765040904292801557711841, 0.971538006563571580067391165211, 3.22597078597820652216111153240, 4.14175799466511726546744984124, 4.57487752427294243846116027652, 5.91032976499424730380844169949, 6.65572881975978979255602177106, 7.65476510772946804913267177056, 8.361723125046059904844572810759, 8.914549704120438996570822207370, 10.12916954783948558106799077791

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