Properties

Label 2-1110-185.184-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.588 - 0.808i$
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  − 2-s + i·3-s + 4-s + (2 + i)5-s i·6-s − 3i·7-s − 8-s − 9-s + (−2 − i)10-s − 3·11-s + i·12-s − 4·13-s + 3i·14-s + (−1 + 2i)15-s + 16-s + 7·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + (0.894 + 0.447i)5-s − 0.408i·6-s − 1.13i·7-s − 0.353·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.904·11-s + 0.288i·12-s − 1.10·13-s + 0.801i·14-s + (−0.258 + 0.516i)15-s + 0.250·16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.588 - 0.808i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270573610\)
\(L(\frac12)\) \(\approx\) \(1.270573610\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
5 \( 1 + (-2 - i)T \)
37 \( 1 + (-6 - i)T \)
good7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 - 5iT - 31T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 5iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.944785342426537184050889714096, −9.523444578773864700398708753408, −8.338064904277218959335980431427, −7.43135631203755985745287464329, −6.92438416930772307260789120583, −5.61658549998341156981163708296, −5.01645746251138040909617075311, −3.51741779286829605121937274790, −2.67749981244277732549187936917, −1.17622194661047059541889486780, 0.824332873162489645089156544143, 2.36715974832215047004468296495, 2.70870410307633660129651567510, 4.88171201275687955830839733343, 5.62898898145511302419375314104, 6.27035422248702251553430672134, 7.56616110415167812477128210763, 7.944800691313305294392262098999, 9.142063290605765694918676228180, 9.464422765222231864240239895120

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