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Properties

Label 2-1110-185.142-c1-0-21
Degree $2$
Conductor $1110$
Sign $-0.481 + 0.876i$
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 + (−0.943 + 2.02i)5-s + (−0.707 + 0.707i)6-s + (−0.516 − 0.516i)7-s + i·8-s + 1.00i·9-s + (2.02 + 0.943i)10-s − 0.497i·11-s + (0.707 + 0.707i)12-s + 1.53i·13-s + (−0.516 + 0.516i)14-s + (2.10 − 0.766i)15-s + 16-s + 2.55·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.421 + 0.906i)5-s + (−0.288 + 0.288i)6-s + (−0.195 − 0.195i)7-s + 0.353i·8-s + 0.333i·9-s + (0.641 + 0.298i)10-s − 0.149i·11-s + (0.204 + 0.204i)12-s + 0.424i·13-s + (−0.137 + 0.137i)14-s + (0.542 − 0.197i)15-s + 0.250·16-s + 0.619·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.481 + 0.876i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.481 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9178284759\)
\(L(\frac12)\) \(\approx\) \(0.9178284759\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.943 - 2.02i)T \)
37 \( 1 + (3.93 - 4.64i)T \)
good7 \( 1 + (0.516 + 0.516i)T + 7iT^{2} \)
11 \( 1 + 0.497iT - 11T^{2} \)
13 \( 1 - 1.53iT - 13T^{2} \)
17 \( 1 - 2.55T + 17T^{2} \)
19 \( 1 + (2.36 + 2.36i)T + 19iT^{2} \)
23 \( 1 + 5.13iT - 23T^{2} \)
29 \( 1 + (-7.56 + 7.56i)T - 29iT^{2} \)
31 \( 1 + (-5.92 - 5.92i)T + 31iT^{2} \)
41 \( 1 + 2.70iT - 41T^{2} \)
43 \( 1 + 9.05iT - 43T^{2} \)
47 \( 1 + (9.26 + 9.26i)T + 47iT^{2} \)
53 \( 1 + (0.413 - 0.413i)T - 53iT^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (-10.7 - 10.7i)T + 61iT^{2} \)
67 \( 1 + (-9.29 + 9.29i)T - 67iT^{2} \)
71 \( 1 + 4.10T + 71T^{2} \)
73 \( 1 + (-2.63 - 2.63i)T + 73iT^{2} \)
79 \( 1 + (-0.827 - 0.827i)T + 79iT^{2} \)
83 \( 1 + (-11.2 + 11.2i)T - 83iT^{2} \)
89 \( 1 + (6.24 - 6.24i)T - 89iT^{2} \)
97 \( 1 - 3.13T + 97T^{2} \)
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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

−10.09052257393665118836943658669, −8.634784619026504720026011581234, −8.062706451450760224446535963330, −6.78684244579415857769126240881, −6.52189385169480908521514238040, −5.13886948405836173806108422895, −4.15954510865332652259287630402, −3.12674008544436784364799108200, −2.16152936144871906277879775928, −0.50586796594217445627634092885, 1.15429662125501255173174369617, 3.20973421919164305051023126003, 4.27090512174189751387244533515, 5.04044804100493593421534703275, 5.79246786590702551332479657729, 6.65550454488436045757770206741, 7.84909473243324784668890203909, 8.274736012020893951591269802079, 9.344506087923582720888632258684, 9.828528817861473702156512552070

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