Properties

Label 2-1110-185.142-c1-0-29
Degree $2$
Conductor $1110$
Sign $-0.0385 + 0.999i$
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 + (−0.220 + 2.22i)5-s + (0.707 − 0.707i)6-s + (−1.18 − 1.18i)7-s i·8-s + 1.00i·9-s + (−2.22 − 0.220i)10-s + 4.79i·11-s + (0.707 + 0.707i)12-s − 4.44i·13-s + (1.18 − 1.18i)14-s + (1.72 − 1.41i)15-s + 16-s − 1.30·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.0987 + 0.995i)5-s + (0.288 − 0.288i)6-s + (−0.448 − 0.448i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.703 − 0.0698i)10-s + 1.44i·11-s + (0.204 + 0.204i)12-s − 1.23i·13-s + (0.316 − 0.316i)14-s + (0.446 − 0.365i)15-s + 0.250·16-s − 0.315·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0385 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0385 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.0385 + 0.999i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.0385 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2943012699\)
\(L(\frac12)\) \(\approx\) \(0.2943012699\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.220 - 2.22i)T \)
37 \( 1 + (-3.31 + 5.10i)T \)
good7 \( 1 + (1.18 + 1.18i)T + 7iT^{2} \)
11 \( 1 - 4.79iT - 11T^{2} \)
13 \( 1 + 4.44iT - 13T^{2} \)
17 \( 1 + 1.30T + 17T^{2} \)
19 \( 1 + (4.46 + 4.46i)T + 19iT^{2} \)
23 \( 1 - 0.933iT - 23T^{2} \)
29 \( 1 + (3.66 - 3.66i)T - 29iT^{2} \)
31 \( 1 + (-4.54 - 4.54i)T + 31iT^{2} \)
41 \( 1 + 7.79iT - 41T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 + (6.81 + 6.81i)T + 47iT^{2} \)
53 \( 1 + (-5.08 + 5.08i)T - 53iT^{2} \)
59 \( 1 + (3.87 + 3.87i)T + 59iT^{2} \)
61 \( 1 + (6.36 + 6.36i)T + 61iT^{2} \)
67 \( 1 + (-0.639 + 0.639i)T - 67iT^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + (1.06 + 1.06i)T + 73iT^{2} \)
79 \( 1 + (2.21 + 2.21i)T + 79iT^{2} \)
83 \( 1 + (6.32 - 6.32i)T - 83iT^{2} \)
89 \( 1 + (-3.11 + 3.11i)T - 89iT^{2} \)
97 \( 1 - 0.0235T + 97T^{2} \)
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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

−9.770664330518859348234473610828, −8.642761691423529028332524085648, −7.61300534963979971685244913232, −7.02080809340228534739424181541, −6.58212141562674271682023970794, −5.51477972287275976326694837783, −4.55713085881649760738210604814, −3.42917036206740453150767168221, −2.16803855450648231687000867541, −0.14009834903949320378543697717, 1.38291540500776449405042379443, 2.82046639027671690133603513918, 4.07731768956973996107317184060, 4.56086008360934674249613571156, 5.92214672161979116844638728930, 6.20018864333863322977650211934, 7.955501978741571975268752987266, 8.663004711101327263553704815903, 9.305968200014201470100772936932, 9.968420932719543865194099691848

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