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Properties

Label 2-1110-185.142-c1-0-27
Degree $2$
Conductor $1110$
Sign $-0.325 + 0.945i$
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 + (−2.06 + 0.866i)5-s + (0.707 − 0.707i)6-s + (−0.662 − 0.662i)7-s + i·8-s + 1.00i·9-s + (0.866 + 2.06i)10-s + 3.51i·11-s + (−0.707 − 0.707i)12-s − 6.87i·13-s + (−0.662 + 0.662i)14-s + (−2.07 − 0.844i)15-s + 16-s − 2.17·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.921 + 0.387i)5-s + (0.288 − 0.288i)6-s + (−0.250 − 0.250i)7-s + 0.353i·8-s + 0.333i·9-s + (0.273 + 0.651i)10-s + 1.06i·11-s + (−0.204 − 0.204i)12-s − 1.90i·13-s + (−0.176 + 0.176i)14-s + (−0.534 − 0.218i)15-s + 0.250·16-s − 0.527·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.325 + 0.945i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.325 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045607019\)
\(L(\frac12)\) \(\approx\) \(1.045607019\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.06 - 0.866i)T \)
37 \( 1 + (1.24 + 5.95i)T \)
good7 \( 1 + (0.662 + 0.662i)T + 7iT^{2} \)
11 \( 1 - 3.51iT - 11T^{2} \)
13 \( 1 + 6.87iT - 13T^{2} \)
17 \( 1 + 2.17T + 17T^{2} \)
19 \( 1 + (-1.01 - 1.01i)T + 19iT^{2} \)
23 \( 1 + 9.57iT - 23T^{2} \)
29 \( 1 + (-6.49 + 6.49i)T - 29iT^{2} \)
31 \( 1 + (-2.38 - 2.38i)T + 31iT^{2} \)
41 \( 1 - 3.47iT - 41T^{2} \)
43 \( 1 + 2.74iT - 43T^{2} \)
47 \( 1 + (-3.59 - 3.59i)T + 47iT^{2} \)
53 \( 1 + (2.93 - 2.93i)T - 53iT^{2} \)
59 \( 1 + (-3.45 - 3.45i)T + 59iT^{2} \)
61 \( 1 + (9.99 + 9.99i)T + 61iT^{2} \)
67 \( 1 + (-1.14 + 1.14i)T - 67iT^{2} \)
71 \( 1 + 7.32T + 71T^{2} \)
73 \( 1 + (-0.292 - 0.292i)T + 73iT^{2} \)
79 \( 1 + (7.95 + 7.95i)T + 79iT^{2} \)
83 \( 1 + (1.46 - 1.46i)T - 83iT^{2} \)
89 \( 1 + (-6.43 + 6.43i)T - 89iT^{2} \)
97 \( 1 - 0.301T + 97T^{2} \)
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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

−9.948753608156544343512298927931, −8.733586925177304057311526668908, −8.086857853525915565989084397577, −7.35019619425989975341409922584, −6.23579358401692105812774823277, −4.82278051113974902680505434159, −4.24049707860916864655268664119, −3.17523998640869354335886748781, −2.47530322000802822706971579008, −0.47943109858464493491704585173, 1.33149023307179151271119857762, 3.06982097827231359825278196762, 3.99785469938828077716273990989, 4.91997743313898048046457872418, 6.06661984320949920738855331664, 6.90427530307665853553135756059, 7.52386771922060004108003299245, 8.590843948584604167722098345798, 8.868782381081846133019088325575, 9.697289876600750144192477220216

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