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Properties

Label 2-1110-111.68-c1-0-19
Degree $2$
Conductor $1110$
Sign $0.789 - 0.614i$
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  + (−0.707 + 0.707i)2-s + (1.70 − 0.278i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.01 + 1.40i)6-s − 0.348·7-s + (0.707 + 0.707i)8-s + (2.84 − 0.953i)9-s − 1.00·10-s − 3.72·11-s + (−0.278 − 1.70i)12-s + (0.586 − 0.586i)13-s + (0.246 − 0.246i)14-s + (1.40 + 1.01i)15-s − 1.00·16-s + (2.90 + 2.90i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.986 − 0.160i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.412 + 0.573i)6-s − 0.131·7-s + (0.250 + 0.250i)8-s + (0.948 − 0.317i)9-s − 0.316·10-s − 1.12·11-s + (−0.0804 − 0.493i)12-s + (0.162 − 0.162i)13-s + (0.0657 − 0.0657i)14-s + (0.363 + 0.261i)15-s − 0.250·16-s + (0.705 + 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.789 - 0.614i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.789 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.901431955\)
\(L(\frac12)\) \(\approx\) \(1.901431955\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.70 + 0.278i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-2.18 + 5.67i)T \)
good7 \( 1 + 0.348T + 7T^{2} \)
11 \( 1 + 3.72T + 11T^{2} \)
13 \( 1 + (-0.586 + 0.586i)T - 13iT^{2} \)
17 \( 1 + (-2.90 - 2.90i)T + 17iT^{2} \)
19 \( 1 + (-3.00 + 3.00i)T - 19iT^{2} \)
23 \( 1 + (-6.48 - 6.48i)T + 23iT^{2} \)
29 \( 1 + (0.289 - 0.289i)T - 29iT^{2} \)
31 \( 1 + (-4.18 - 4.18i)T + 31iT^{2} \)
41 \( 1 - 9.94T + 41T^{2} \)
43 \( 1 + (4.12 - 4.12i)T - 43iT^{2} \)
47 \( 1 + 7.55iT - 47T^{2} \)
53 \( 1 + 0.164iT - 53T^{2} \)
59 \( 1 + (-6.51 - 6.51i)T + 59iT^{2} \)
61 \( 1 + (8.60 + 8.60i)T + 61iT^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 - 0.738iT - 71T^{2} \)
73 \( 1 + 15.9iT - 73T^{2} \)
79 \( 1 + (-6.84 + 6.84i)T - 79iT^{2} \)
83 \( 1 - 9.66iT - 83T^{2} \)
89 \( 1 + (5.63 - 5.63i)T - 89iT^{2} \)
97 \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \)
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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

−9.713133390617963557624443033605, −9.115584095534850377035736257840, −8.184138158580955356458535941978, −7.55463334870911202407391402983, −6.90120646192909642401177921239, −5.78755222277582341579585125625, −4.91549114551398492450909406787, −3.44050133512563370176671766373, −2.62644064198926165075380705687, −1.26122788108025664120456619502, 1.09358443499634929632054395121, 2.51359852854107790287290278726, 3.10142463206802102074990481658, 4.38317052874155210952718600928, 5.26437074557567837708783523302, 6.61870301741451689024576285512, 7.72309691840129571589072088959, 8.122117270745368466734929883517, 9.094469657695495842943785735778, 9.692309917927694412733531773374

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