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Properties

Label 2-1110-185.84-c1-0-26
Degree $2$
Conductor $1110$
Sign $0.993 + 0.117i$
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 + (0.529 − 2.17i)5-s + 0.999·6-s + (−2.93 + 1.69i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.54 − 1.61i)10-s + 4.29·11-s + (0.866 + 0.499i)12-s + (1.80 − 1.04i)13-s − 3.38·14-s + (−0.627 − 2.14i)15-s + (−0.5 + 0.866i)16-s + (5.53 + 3.19i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.236 − 0.971i)5-s + 0.408·6-s + (−1.10 + 0.640i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.488 − 0.511i)10-s + 1.29·11-s + (0.249 + 0.144i)12-s + (0.501 − 0.289i)13-s − 0.905·14-s + (−0.162 − 0.554i)15-s + (−0.125 + 0.216i)16-s + (1.34 + 0.774i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.993 + 0.117i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.993 + 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.846220190\)
\(L(\frac12)\) \(\approx\) \(2.846220190\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.529 + 2.17i)T \)
37 \( 1 + (-0.158 + 6.08i)T \)
good7 \( 1 + (2.93 - 1.69i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 + (-1.80 + 1.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.53 - 3.19i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.07 + 1.85i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.17iT - 23T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 - 8.38T + 31T^{2} \)
41 \( 1 + (5.28 + 9.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.54iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 + (-9.69 - 5.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.78 - 3.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.89 + 3.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.4 - 7.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.72 + 8.18i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.65iT - 73T^{2} \)
79 \( 1 + (3.37 + 5.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.0 + 5.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.462 - 0.800i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.99iT - 97T^{2} \)
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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

−9.587457023686366277805336750382, −8.831022078342854600831453286716, −8.370568205421792111646261249033, −7.22686432873732231126961146512, −6.12756679009655295545993329948, −5.93117856695766227479177347362, −4.52470408623492885868754554251, −3.67082917907822763751411246284, −2.66925461971977058727619517177, −1.20036584972550291158405908105, 1.41430291950580578118594709127, 2.99595799249131010825442961411, 3.44502914125424126868159549623, 4.27615813307628055092426164015, 5.67350986296167114953124489146, 6.60635271217129566849156061593, 6.99888994883979521203917642954, 8.213622895221106121251409677182, 9.446436370477374530757958358355, 9.966906989435200085887612154054

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