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Properties

Label 2-1110-185.43-c1-0-3
Degree $2$
Conductor $1110$
Sign $0.256 - 0.966i$
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 + (0.215 + 2.22i)5-s + (−0.707 − 0.707i)6-s + (−1.35 + 1.35i)7-s + i·8-s − 1.00i·9-s + (2.22 − 0.215i)10-s + 0.0194i·11-s + (−0.707 + 0.707i)12-s − 0.159i·13-s + (1.35 + 1.35i)14-s + (1.72 + 1.42i)15-s + 16-s − 4.04·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.0961 + 0.995i)5-s + (−0.288 − 0.288i)6-s + (−0.512 + 0.512i)7-s + 0.353i·8-s − 0.333i·9-s + (0.703 − 0.0679i)10-s + 0.00585i·11-s + (−0.204 + 0.204i)12-s − 0.0441i·13-s + (0.362 + 0.362i)14-s + (0.445 + 0.367i)15-s + 0.250·16-s − 0.979·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.256 - 0.966i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.256 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9447452252\)
\(L(\frac12)\) \(\approx\) \(0.9447452252\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.215 - 2.22i)T \)
37 \( 1 + (-1.68 - 5.84i)T \)
good7 \( 1 + (1.35 - 1.35i)T - 7iT^{2} \)
11 \( 1 - 0.0194iT - 11T^{2} \)
13 \( 1 + 0.159iT - 13T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
19 \( 1 + (4.58 - 4.58i)T - 19iT^{2} \)
23 \( 1 - 2.53iT - 23T^{2} \)
29 \( 1 + (0.812 + 0.812i)T + 29iT^{2} \)
31 \( 1 + (-2.35 + 2.35i)T - 31iT^{2} \)
41 \( 1 - 8.21iT - 41T^{2} \)
43 \( 1 - 4.36iT - 43T^{2} \)
47 \( 1 + (5.16 - 5.16i)T - 47iT^{2} \)
53 \( 1 + (4.20 + 4.20i)T + 53iT^{2} \)
59 \( 1 + (3.16 - 3.16i)T - 59iT^{2} \)
61 \( 1 + (-1.82 + 1.82i)T - 61iT^{2} \)
67 \( 1 + (2.18 + 2.18i)T + 67iT^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 + (-5.46 + 5.46i)T - 73iT^{2} \)
79 \( 1 + (-4.93 + 4.93i)T - 79iT^{2} \)
83 \( 1 + (-6.36 - 6.36i)T + 83iT^{2} \)
89 \( 1 + (10.2 + 10.2i)T + 89iT^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−9.925707207709868864078009061709, −9.435084386632676162721886179879, −8.383110537236712917897513986820, −7.70514470325888160613011879810, −6.45475262824977175017781154413, −6.11070535988612700898514046523, −4.57585015101331097251666611900, −3.47791823811860069419754837610, −2.69543773925456840836317113287, −1.78384749988718867804618568238, 0.37488203383974855548177457953, 2.23670713448363745466661659231, 3.77158113135158229545207472019, 4.49163202642843920349362909530, 5.26692824342268761591815746076, 6.41836167120440253294958186575, 7.11211765875372977890991522199, 8.220257759576332194692885890307, 8.843145639611939401798972312631, 9.357246306921689491129112466676

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