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Properties

Label 2-1110-185.117-c1-0-23
Degree $2$
Conductor $1110$
Sign $0.984 - 0.176i$
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 + (0.707 + 0.707i)3-s + 4-s + (−0.161 − 2.23i)5-s + (0.707 + 0.707i)6-s + (1.80 + 1.80i)7-s + 8-s + 1.00i·9-s + (−0.161 − 2.23i)10-s + 1.06i·11-s + (0.707 + 0.707i)12-s + 5.30·13-s + (1.80 + 1.80i)14-s + (1.46 − 1.69i)15-s + 16-s − 0.822i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.0721 − 0.997i)5-s + (0.288 + 0.288i)6-s + (0.682 + 0.682i)7-s + 0.353·8-s + 0.333i·9-s + (−0.0510 − 0.705i)10-s + 0.320i·11-s + (0.204 + 0.204i)12-s + 1.47·13-s + (0.482 + 0.482i)14-s + (0.377 − 0.436i)15-s + 0.250·16-s − 0.199i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.984 - 0.176i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.984 - 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.196678683\)
\(L(\frac12)\) \(\approx\) \(3.196678683\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.161 + 2.23i)T \)
37 \( 1 + (-5.68 - 2.17i)T \)
good7 \( 1 + (-1.80 - 1.80i)T + 7iT^{2} \)
11 \( 1 - 1.06iT - 11T^{2} \)
13 \( 1 - 5.30T + 13T^{2} \)
17 \( 1 + 0.822iT - 17T^{2} \)
19 \( 1 + (0.279 - 0.279i)T - 19iT^{2} \)
23 \( 1 + 6.71T + 23T^{2} \)
29 \( 1 + (1.25 + 1.25i)T + 29iT^{2} \)
31 \( 1 + (-3.59 + 3.59i)T - 31iT^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 - 9.11T + 43T^{2} \)
47 \( 1 + (-6.22 - 6.22i)T + 47iT^{2} \)
53 \( 1 + (-1.25 + 1.25i)T - 53iT^{2} \)
59 \( 1 + (10.0 - 10.0i)T - 59iT^{2} \)
61 \( 1 + (-5.97 + 5.97i)T - 61iT^{2} \)
67 \( 1 + (2.68 - 2.68i)T - 67iT^{2} \)
71 \( 1 + 6.33T + 71T^{2} \)
73 \( 1 + (9.12 + 9.12i)T + 73iT^{2} \)
79 \( 1 + (10.8 - 10.8i)T - 79iT^{2} \)
83 \( 1 + (2.66 - 2.66i)T - 83iT^{2} \)
89 \( 1 + (1.96 + 1.96i)T + 89iT^{2} \)
97 \( 1 + 2.76iT - 97T^{2} \)
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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

−9.775600569368074709207266166514, −8.937846028233104655238243633900, −8.269043483289656594440297478598, −7.61016458153832492097551730962, −6.05564763142648830644098386266, −5.60473131276047460425824770603, −4.42563767788449613627313918186, −4.03168105850047701133725311295, −2.58706887682965810602826852823, −1.46473964651948723925206364328, 1.38837416106572396229921351556, 2.61422126974783015280442152360, 3.67534554919079678043005714094, 4.27447827442048465483553651166, 5.82102176924821103442940137500, 6.34817866220303426751638262420, 7.34212899952557119736470313367, 7.935969383255614689439507101348, 8.787368734235583631931722222095, 10.10620275228886291587674570987

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