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Properties

Label 2-1110-37.10-c1-0-24
Degree $2$
Conductor $1110$
Sign $-0.991 + 0.131i$
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 + (−2.44 − 4.22i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + 2.56·11-s + (−0.499 + 0.866i)12-s + (−1.71 − 2.96i)13-s + 4.88·14-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (1.90 − 3.30i)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 + (−0.922 − 1.59i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s + 0.773·11-s + (−0.144 + 0.249i)12-s + (−0.475 − 0.822i)13-s + 1.30·14-s + (−0.129 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.462 − 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.991 + 0.131i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.991 + 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4321035382\)
\(L(\frac12)\) \(\approx\) \(0.4321035382\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.11 - 5.22i)T \)
good7 \( 1 + (2.44 + 4.22i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 + (1.71 + 2.96i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.90 + 3.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.76 - 3.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.27T + 23T^{2} \)
29 \( 1 - 6.98T + 29T^{2} \)
31 \( 1 + 7.03T + 31T^{2} \)
41 \( 1 + (3.87 + 6.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.73T + 43T^{2} \)
47 \( 1 + 7.62T + 47T^{2} \)
53 \( 1 + (-0.554 + 0.960i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.20 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.12 + 7.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.59 - 9.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0742 + 0.128i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.89T + 73T^{2} \)
79 \( 1 + (-1.61 - 2.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.63 + 4.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.96 - 3.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−9.601813762715816555083410272516, −8.409390667942818044630900483349, −7.58260616501747830407043660500, −7.08717534974459054776951158227, −6.28879928977860126086560934779, −5.33120273246331966433777623447, −4.22958262953096246950109231628, −3.27629351838599239376051851841, −1.28833881508080984965863859944, −0.23977084562860641030283364955, 1.95769296861225970550583339003, 3.03568498960165567414273994150, 3.86258222472288870529499529801, 5.02762858235656732639496271225, 6.10733870478154170435864169467, 6.71518851339981185646374789581, 8.012337356176317763429802368703, 9.000133289395797816072128408707, 9.400749124185877916293832884316, 10.10222083124435754938806938458

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