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Properties

Label 2-1110-185.43-c1-0-30
Degree $2$
Conductor $1110$
Sign $-0.271 + 0.962i$
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  − i·2-s + (0.707 − 0.707i)3-s − 4-s + (2.04 + 0.906i)5-s + (−0.707 − 0.707i)6-s + (1.03 − 1.03i)7-s + i·8-s − 1.00i·9-s + (0.906 − 2.04i)10-s − 6.26i·11-s + (−0.707 + 0.707i)12-s + 0.736i·13-s + (−1.03 − 1.03i)14-s + (2.08 − 0.804i)15-s + 16-s + 2.79·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.914 + 0.405i)5-s + (−0.288 − 0.288i)6-s + (0.390 − 0.390i)7-s + 0.353i·8-s − 0.333i·9-s + (0.286 − 0.646i)10-s − 1.88i·11-s + (−0.204 + 0.204i)12-s + 0.204i·13-s + (−0.276 − 0.276i)14-s + (0.538 − 0.207i)15-s + 0.250·16-s + 0.677·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.271 + 0.962i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.140990040\)
\(L(\frac12)\) \(\approx\) \(2.140990040\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-2.04 - 0.906i)T \)
37 \( 1 + (-4.32 + 4.27i)T \)
good7 \( 1 + (-1.03 + 1.03i)T - 7iT^{2} \)
11 \( 1 + 6.26iT - 11T^{2} \)
13 \( 1 - 0.736iT - 13T^{2} \)
17 \( 1 - 2.79T + 17T^{2} \)
19 \( 1 + (0.886 - 0.886i)T - 19iT^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 + (5.78 + 5.78i)T + 29iT^{2} \)
31 \( 1 + (-1.64 + 1.64i)T - 31iT^{2} \)
41 \( 1 - 0.847iT - 41T^{2} \)
43 \( 1 + 9.33iT - 43T^{2} \)
47 \( 1 + (-2.86 + 2.86i)T - 47iT^{2} \)
53 \( 1 + (-3.98 - 3.98i)T + 53iT^{2} \)
59 \( 1 + (-8.40 + 8.40i)T - 59iT^{2} \)
61 \( 1 + (5.05 - 5.05i)T - 61iT^{2} \)
67 \( 1 + (6.24 + 6.24i)T + 67iT^{2} \)
71 \( 1 + 0.962T + 71T^{2} \)
73 \( 1 + (2.59 - 2.59i)T - 73iT^{2} \)
79 \( 1 + (11.6 - 11.6i)T - 79iT^{2} \)
83 \( 1 + (7.04 + 7.04i)T + 83iT^{2} \)
89 \( 1 + (-12.7 - 12.7i)T + 89iT^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−9.610931582595674600537593208216, −8.935238625853807819223016191649, −8.050675139380744641430469122220, −7.25986191562002374578724917942, −5.93940286431066257147362140666, −5.57988549388117120458068387342, −3.95421305098407797090369068000, −3.17590164797038011510740947164, −2.11069334831515224385040143283, −0.983327227289698627976515500696, 1.63961860358783674641929968538, 2.74438040158919496485604756706, 4.37697709539958938844513201719, 4.89215399996386371929472257251, 5.76094660870390697060000626767, 6.78684486046643982798366485848, 7.61461130075080705664378230203, 8.558300405117567012930783845197, 9.163559010026090775939891409173, 9.983858197491482904549725399891

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