Properties

Label 2-1110-185.84-c1-0-16
Degree $2$
Conductor $1110$
Sign $0.999 + 0.0267i$
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 + (2.22 + 0.176i)5-s − 0.999·6-s + (−3.02 + 1.74i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.84 − 1.26i)10-s + 5.35·11-s + (0.866 + 0.499i)12-s + (−2.41 + 1.39i)13-s + 3.48·14-s + (2.01 − 0.961i)15-s + (−0.5 + 0.866i)16-s + (0.818 + 0.472i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.996 + 0.0789i)5-s − 0.408·6-s + (−1.14 + 0.659i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.582 − 0.400i)10-s + 1.61·11-s + (0.249 + 0.144i)12-s + (−0.669 + 0.386i)13-s + 0.932·14-s + (0.521 − 0.248i)15-s + (−0.125 + 0.216i)16-s + (0.198 + 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.0267i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.999 + 0.0267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.636017908\)
\(L(\frac12)\) \(\approx\) \(1.636017908\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.176i)T \)
37 \( 1 + (-6.05 - 0.576i)T \)
good7 \( 1 + (3.02 - 1.74i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.35T + 11T^{2} \)
13 \( 1 + (2.41 - 1.39i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.818 - 0.472i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.113 + 0.195i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.422iT - 23T^{2} \)
29 \( 1 + 2.54T + 29T^{2} \)
31 \( 1 - 9.36T + 31T^{2} \)
41 \( 1 + (-4.96 - 8.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.646iT - 43T^{2} \)
47 \( 1 + 2.57iT - 47T^{2} \)
53 \( 1 + (1.84 + 1.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.21 - 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.10 - 1.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.10 + 3.52i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0551 + 0.0955i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.85iT - 73T^{2} \)
79 \( 1 + (8.57 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.732 - 0.423i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.12 + 8.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.24iT - 97T^{2} \)
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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

−9.514724128957773274572706376165, −9.342562953591073159857374971383, −8.517806537829064897985963687898, −7.32703388176403252852033656374, −6.43003971836038776276328896565, −6.06543214757365948303757190962, −4.43484104286495820944235922259, −3.18384311145511464870418049950, −2.41855119193194626430412471949, −1.27095820013497695990362227468, 0.983974880281472588684611079262, 2.40706700639910291063818693208, 3.53140167511938695535394750318, 4.63491614021749932575502782938, 5.91936616039031478029774870556, 6.56280304876975703033076310971, 7.27575782278625609710599856278, 8.360280400836957717628755435950, 9.366006516092709845142039782341, 9.599623120693159130991253969646

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