Properties

Label 2-1110-37.11-c1-0-19
Degree $2$
Conductor $1110$
Sign $0.629 + 0.777i$
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  + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·6-s + (1.89 − 3.28i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 3.84·11-s + (0.499 + 0.866i)12-s + (5.82 + 3.36i)13-s − 3.78i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (4.14 − 2.39i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (0.715 − 1.23i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.15·11-s + (0.144 + 0.249i)12-s + (1.61 + 0.932i)13-s − 1.01i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (1.00 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.629 + 0.777i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.629 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.386887441\)
\(L(\frac12)\) \(\approx\) \(2.386887441\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-2.29 - 5.63i)T \)
good7 \( 1 + (-1.89 + 3.28i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.84T + 11T^{2} \)
13 \( 1 + (-5.82 - 3.36i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.14 + 2.39i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.79 + 2.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.45iT - 23T^{2} \)
29 \( 1 + 1.86iT - 29T^{2} \)
31 \( 1 - 8.49iT - 31T^{2} \)
41 \( 1 + (-4.99 + 8.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.0397iT - 43T^{2} \)
47 \( 1 - 1.04T + 47T^{2} \)
53 \( 1 + (1.60 + 2.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.56 + 0.902i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.58 + 7.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.928 - 1.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 + (3.13 + 1.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.79 - 11.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-14.8 + 8.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.15iT - 97T^{2} \)
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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

−10.23186233856283226071653385599, −8.961338688969667570448279837593, −8.134184902704045985470206587695, −6.97103159953830113831567470789, −6.29581452944118063071701471638, −5.19255265144755295424748908991, −4.48153156468493188952049907565, −3.68493312002592554316942893608, −2.44104215565122139786347283566, −0.993130458707182892315127290757, 1.53124206254629286973398831867, 2.61507549061848882803960574565, 3.82223506617132766537714106131, 5.22342299306108722001179183540, 5.79628153173171957772622267567, 6.06614527237441064927399429582, 7.74917204100502922246451680228, 8.030789611724284809357914835955, 8.836149669904354763092702701367, 10.05623968730752048799031865523

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