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Properties

Label 2-1110-37.10-c1-0-20
Degree $2$
Conductor $1110$
Sign $-0.972 + 0.234i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (−1.68 − 2.92i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + 4.37·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s − 3.37·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (3 − 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (−0.637 − 1.10i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s + 1.31·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s − 0.901·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.972 + 0.234i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.367297496\)
\(L(\frac12)\) \(\approx\) \(1.367297496\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (2.55 - 5.51i)T \)
good7 \( 1 + (1.68 + 2.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.18 + 5.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 + 8.11T + 31T^{2} \)
41 \( 1 + (4.37 + 7.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 6.11T + 43T^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
53 \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.18 + 3.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.68 + 8.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.37 - 2.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.11T + 73T^{2} \)
79 \( 1 + (0.0584 + 0.101i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.74 + 9.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.558 - 0.967i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−9.426272422784066659134740805498, −9.069377231658410473148968556321, −7.39567481661882850059375197106, −6.96726428447407423518085692492, −6.17545599783337158385751155978, −5.08094175018132647945746660643, −3.97276363144708872972146069774, −3.20238884400745860649102043135, −1.84464081723413483553197368337, −0.55822790667070444982146460398, 1.79398152111434136095456890858, 3.50525203871378483555146796037, 4.02151685281758240390219948833, 5.44201385356941261467740275265, 5.86068786626288978992305006018, 6.52979496259404026868994716283, 7.77761323328767499486746865246, 8.773144314190925337442852963911, 9.203853288516907286813531526752, 10.01961615866825953939899761784

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