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Properties

Label 2-1110-185.43-c1-0-35
Degree $2$
Conductor $1110$
Sign $-0.951 + 0.308i$
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 + (−0.132 − 2.23i)5-s + (−0.707 − 0.707i)6-s + (1.47 − 1.47i)7-s + i·8-s − 1.00i·9-s + (−2.23 + 0.132i)10-s + 1.15i·11-s + (−0.707 + 0.707i)12-s − 2.88i·13-s + (−1.47 − 1.47i)14-s + (−1.67 − 1.48i)15-s + 16-s + 5.62·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−0.0594 − 0.998i)5-s + (−0.288 − 0.288i)6-s + (0.559 − 0.559i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.705 + 0.0420i)10-s + 0.347i·11-s + (−0.204 + 0.204i)12-s − 0.801i·13-s + (−0.395 − 0.395i)14-s + (−0.431 − 0.383i)15-s + 0.250·16-s + 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.951 + 0.308i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.739093934\)
\(L(\frac12)\) \(\approx\) \(1.739093934\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.132 + 2.23i)T \)
37 \( 1 + (3.82 - 4.72i)T \)
good7 \( 1 + (-1.47 + 1.47i)T - 7iT^{2} \)
11 \( 1 - 1.15iT - 11T^{2} \)
13 \( 1 + 2.88iT - 13T^{2} \)
17 \( 1 - 5.62T + 17T^{2} \)
19 \( 1 + (-1.32 + 1.32i)T - 19iT^{2} \)
23 \( 1 + 2.70iT - 23T^{2} \)
29 \( 1 + (6.59 + 6.59i)T + 29iT^{2} \)
31 \( 1 + (0.920 - 0.920i)T - 31iT^{2} \)
41 \( 1 - 6.84iT - 41T^{2} \)
43 \( 1 - 8.92iT - 43T^{2} \)
47 \( 1 + (-5.84 + 5.84i)T - 47iT^{2} \)
53 \( 1 + (4.27 + 4.27i)T + 53iT^{2} \)
59 \( 1 + (-7.37 + 7.37i)T - 59iT^{2} \)
61 \( 1 + (-0.721 + 0.721i)T - 61iT^{2} \)
67 \( 1 + (-4.32 - 4.32i)T + 67iT^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 + (0.815 - 0.815i)T - 73iT^{2} \)
79 \( 1 + (11.8 - 11.8i)T - 79iT^{2} \)
83 \( 1 + (5.88 + 5.88i)T + 83iT^{2} \)
89 \( 1 + (6.87 + 6.87i)T + 89iT^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−9.693558723972868104814139006127, −8.549702071986846580497970890227, −8.001863187824250196025809339765, −7.32806120091308169052224591291, −5.85328663718030690300502857920, −4.98891857768660388977912817351, −4.10876887728674728896136934015, −3.07169137644568365771837908938, −1.72873832187421907438251868633, −0.77197758933676560987498172411, 1.89461890709459785770724128178, 3.26132129784661501104342297263, 3.98021968868542106785384851753, 5.37139181417207027940915826860, 5.81617818262908796256837048032, 7.18027270654197082147699358135, 7.52113042255396402463497978491, 8.609698266713693277060980169393, 9.218079114864031954140343262326, 10.10976381610235135171885144427

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