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Properties

Label 2-1110-37.26-c1-0-24
Degree $2$
Conductor $1110$
Sign $0.729 + 0.683i$
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 + (2.5 − 4.33i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s + 2·11-s + (0.499 + 0.866i)12-s + (−2.5 + 4.33i)13-s + 5·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)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 + (0.944 − 1.63i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s + 0.603·11-s + (0.144 + 0.249i)12-s + (−0.693 + 1.20i)13-s + 1.33·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.729 + 0.683i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.291591294\)
\(L(\frac12)\) \(\approx\) \(2.291591294\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 6.06i)T \)
good7 \( 1 + (-2.5 + 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + (-4 - 6.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.5 - 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−9.350636411152087656773196620287, −8.991774535737113864566293787271, −7.67836108611901298635361204773, −7.32663558808767185092253249957, −6.67289225518721656527527585044, −5.40896512500361743712494875341, −4.44348654643401444257823562430, −3.91568696558912322202087051379, −2.22617306963070611371022699356, −0.921361052193918108075689182150, 1.75047599663205583719936389376, 2.64771732596061523796186109014, 3.56268178767474618850704171106, 4.81426163624664657202230537606, 5.50005982686468226758144248732, 6.19557285676697858922175379338, 7.78069348500159354035072346839, 8.368236106500410927382404112243, 9.387715447092821216452960063542, 9.831343566594205917664549654888

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