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Properties

Label 2-110-5.4-c3-0-10
Degree $2$
Conductor $110$
Sign $-0.662 + 0.749i$
Analytic cond. $6.49021$
Root an. cond. $2.54758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 6.82i·3-s − 4·4-s + (−8.37 − 7.40i)5-s + 13.6·6-s − 13.6i·7-s + 8i·8-s − 19.5·9-s + (−14.8 + 16.7i)10-s − 11·11-s − 27.2i·12-s − 66.9i·13-s − 27.2·14-s + (50.5 − 57.1i)15-s + 16·16-s − 136. i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.31i·3-s − 0.5·4-s + (−0.749 − 0.662i)5-s + 0.928·6-s − 0.736i·7-s + 0.353i·8-s − 0.723·9-s + (−0.468 + 0.529i)10-s − 0.301·11-s − 0.656i·12-s − 1.42i·13-s − 0.520·14-s + (0.869 − 0.983i)15-s + 0.250·16-s − 1.95i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(6.49021\)
Root analytic conductor: \(2.54758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :3/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.301836 - 0.669950i\)
\(L(\frac12)\) \(\approx\) \(0.301836 - 0.669950i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (8.37 + 7.40i)T \)
11 \( 1 + 11T \)
good3 \( 1 - 6.82iT - 27T^{2} \)
7 \( 1 + 13.6iT - 343T^{2} \)
13 \( 1 + 66.9iT - 2.19e3T^{2} \)
17 \( 1 + 136. iT - 4.91e3T^{2} \)
19 \( 1 + 126.T + 6.85e3T^{2} \)
23 \( 1 - 62.5iT - 1.21e4T^{2} \)
29 \( 1 + 144.T + 2.43e4T^{2} \)
31 \( 1 - 229.T + 2.97e4T^{2} \)
37 \( 1 + 6.56iT - 5.06e4T^{2} \)
41 \( 1 + 276.T + 6.89e4T^{2} \)
43 \( 1 + 233. iT - 7.95e4T^{2} \)
47 \( 1 - 381. iT - 1.03e5T^{2} \)
53 \( 1 + 182. iT - 1.48e5T^{2} \)
59 \( 1 + 111.T + 2.05e5T^{2} \)
61 \( 1 - 116.T + 2.26e5T^{2} \)
67 \( 1 + 480. iT - 3.00e5T^{2} \)
71 \( 1 + 602.T + 3.57e5T^{2} \)
73 \( 1 - 622. iT - 3.89e5T^{2} \)
79 \( 1 - 768.T + 4.93e5T^{2} \)
83 \( 1 + 953. iT - 5.71e5T^{2} \)
89 \( 1 - 757.T + 7.04e5T^{2} \)
97 \( 1 + 1.11e3iT - 9.12e5T^{2} \)
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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

−12.72226596640337241352028424867, −11.51189308958403259151014210833, −10.62888604118241817271227654951, −9.837154319220675796215232379867, −8.750746299560332971525630213845, −7.56006346583797721513148257808, −5.20231228329217200282836601158, −4.36417644245579532387535135971, −3.22392619478303466255590899575, −0.39742227369925742370582013549, 2.08353745027255629824845475999, 4.17055308184002242806594899116, 6.22606896849344642622255523480, 6.73523821168346963236774683911, 7.997941715485798357060330942670, 8.672266589184963061986179000429, 10.49858648181880787506225399030, 11.81509576950247482707799403539, 12.58293138604942079184315460737, 13.53319111451302412198105569292

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