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Properties

Label 2-1110-37.26-c1-0-11
Degree $2$
Conductor $1110$
Sign $0.995 - 0.0915i$
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 + (−0.301 + 0.522i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 4.84·11-s + (−0.499 − 0.866i)12-s + (3.08 − 5.33i)13-s + 0.603·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1.24 − 2.16i)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 + (−0.113 + 0.197i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 1.46·11-s + (−0.144 − 0.249i)12-s + (0.854 − 1.48i)13-s + 0.161·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.302 − 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.995 - 0.0915i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.995 - 0.0915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.186487075\)
\(L(\frac12)\) \(\approx\) \(1.186487075\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.20 - 4.39i)T \)
good7 \( 1 + (0.301 - 0.522i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.84T + 11T^{2} \)
13 \( 1 + (-3.08 + 5.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.24 + 2.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.91 - 6.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.26T + 23T^{2} \)
29 \( 1 - 0.941T + 29T^{2} \)
31 \( 1 - 4.07T + 31T^{2} \)
41 \( 1 + (-4.20 + 7.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 7.28T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + (0.959 + 1.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.737 - 1.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.88 - 8.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.84 - 4.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.34 + 5.78i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (5.70 - 9.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.50 - 2.60i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.949 + 1.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.28T + 97T^{2} \)
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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

−10.04667856122327479556205979235, −9.124548698546904254999621027105, −8.415603702759330471699736447211, −7.56723949620979165269669227605, −6.29468886487918310465113701451, −5.76439840344989516432700251399, −4.20250128059161856853331009837, −3.74016857330862992747955734930, −2.56513083541861810257730969324, −0.987263270012034434811042085503, 0.863914775106318183986176893240, 2.06151730657264172038110859981, 4.05335786890868283276529989348, 4.48444205460281475305658033677, 6.04533824647858919047026033042, 6.47363504115472055018154938145, 7.17185851742671631856906923437, 8.248047222434176797157444555955, 9.002464825116403583844816700039, 9.405975919808287319360300954058

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