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Properties

Label 2-1110-185.174-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.283 - 0.959i$
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 + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (1.61 + 1.54i)5-s − 0.999·6-s + (1.82 + 1.05i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.17 − 0.527i)10-s + 1.56·11-s + (0.866 − 0.499i)12-s + (0.735 + 0.424i)13-s − 2.11·14-s + (0.629 + 2.14i)15-s + (−0.5 − 0.866i)16-s + (5.36 − 3.09i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.723 + 0.690i)5-s − 0.408·6-s + (0.691 + 0.399i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.687 − 0.166i)10-s + 0.472·11-s + (0.249 − 0.144i)12-s + (0.204 + 0.117i)13-s − 0.564·14-s + (0.162 + 0.553i)15-s + (−0.125 − 0.216i)16-s + (1.30 − 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.283 - 0.959i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.283 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.885569397\)
\(L(\frac12)\) \(\approx\) \(1.885569397\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-1.61 - 1.54i)T \)
37 \( 1 + (3.54 + 4.94i)T \)
good7 \( 1 + (-1.82 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + (-0.735 - 0.424i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.36 + 3.09i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.926 - 1.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.02iT - 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
41 \( 1 + (1.71 - 2.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.80iT - 43T^{2} \)
47 \( 1 + 8.42iT - 47T^{2} \)
53 \( 1 + (-7.19 + 4.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.267 - 0.463i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.39 - 9.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0321 + 0.0185i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.11 - 8.85i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.96iT - 73T^{2} \)
79 \( 1 + (-4.21 + 7.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.938 + 0.542i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.64 + 8.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−9.965083365689425611522079232752, −9.136996168487448488793567272062, −8.500788882657887570191703716790, −7.57181049174864390049450708430, −6.84047297381103619482811617649, −5.80563750473698851206737727386, −5.09848884161430282398731489770, −3.68067584793744870129698109953, −2.54104472157376519981100511616, −1.50794166988349840578781710339, 1.14115413469902944795703515781, 1.83591391575454400884009236821, 3.21968707007715831668373489227, 4.29638948675084759438912328038, 5.42539405853373303465634548334, 6.41813058607954462500334582856, 7.48776944314288649988337806514, 8.182940416306736816983333961608, 8.863273062755927059103057566626, 9.587266808214552777919765892018

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