Properties

Label 2-1110-185.68-c1-0-19
Degree $2$
Conductor $1110$
Sign $0.946 + 0.322i$
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 + (−0.707 + 0.707i)3-s + 4-s + (2.23 + 0.0221i)5-s + (0.707 − 0.707i)6-s + (−0.281 + 0.281i)7-s − 8-s − 1.00i·9-s + (−2.23 − 0.0221i)10-s − 1.64i·11-s + (−0.707 + 0.707i)12-s + 3.15·13-s + (0.281 − 0.281i)14-s + (−1.59 + 1.56i)15-s + 16-s − 3.44i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (0.999 + 0.00990i)5-s + (0.288 − 0.288i)6-s + (−0.106 + 0.106i)7-s − 0.353·8-s − 0.333i·9-s + (−0.707 − 0.00700i)10-s − 0.496i·11-s + (−0.204 + 0.204i)12-s + 0.875·13-s + (0.0752 − 0.0752i)14-s + (−0.412 + 0.404i)15-s + 0.250·16-s − 0.835i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.946 + 0.322i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210409094\)
\(L(\frac12)\) \(\approx\) \(1.210409094\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-2.23 - 0.0221i)T \)
37 \( 1 + (-0.925 - 6.01i)T \)
good7 \( 1 + (0.281 - 0.281i)T - 7iT^{2} \)
11 \( 1 + 1.64iT - 11T^{2} \)
13 \( 1 - 3.15T + 13T^{2} \)
17 \( 1 + 3.44iT - 17T^{2} \)
19 \( 1 + (1.85 + 1.85i)T + 19iT^{2} \)
23 \( 1 - 3.91T + 23T^{2} \)
29 \( 1 + (-2.67 + 2.67i)T - 29iT^{2} \)
31 \( 1 + (6.77 + 6.77i)T + 31iT^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 - 1.27T + 43T^{2} \)
47 \( 1 + (0.752 - 0.752i)T - 47iT^{2} \)
53 \( 1 + (-3.88 - 3.88i)T + 53iT^{2} \)
59 \( 1 + (6.53 + 6.53i)T + 59iT^{2} \)
61 \( 1 + (0.144 + 0.144i)T + 61iT^{2} \)
67 \( 1 + (0.492 + 0.492i)T + 67iT^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (-0.804 + 0.804i)T - 73iT^{2} \)
79 \( 1 + (-2.86 - 2.86i)T + 79iT^{2} \)
83 \( 1 + (-3.45 - 3.45i)T + 83iT^{2} \)
89 \( 1 + (-12.5 + 12.5i)T - 89iT^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.609945017177219240879962253479, −9.204594838817826378947484874135, −8.425559018114580560843197224291, −7.25467028924830245720259469829, −6.33817494168126018361569654094, −5.76296358294706418628172785321, −4.76978354533832239905396824721, −3.38446007639234779135628521154, −2.26620678925808932095748352347, −0.822403670398993610246555534925, 1.22806458449160464848578474904, 2.09589553875311115649236893942, 3.47772224080578416927232331452, 4.95197309075908995615778456038, 5.90805757984863132189347017170, 6.55786652449716042070282305626, 7.30317999804998258143115947119, 8.417990784278047273570661149597, 9.013994536928556591861075006968, 9.947872280262607292763614453787

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