Properties

Label 2-1110-185.117-c1-0-0
Degree $2$
Conductor $1110$
Sign $-0.995 - 0.0935i$
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  − 2-s + (−0.707 − 0.707i)3-s + 4-s + (−2.23 − 0.0484i)5-s + (0.707 + 0.707i)6-s + (2.77 + 2.77i)7-s − 8-s + 1.00i·9-s + (2.23 + 0.0484i)10-s + 2.74i·11-s + (−0.707 − 0.707i)12-s − 2.09·13-s + (−2.77 − 2.77i)14-s + (1.54 + 1.61i)15-s + 16-s − 4.54i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (−0.999 − 0.0216i)5-s + (0.288 + 0.288i)6-s + (1.04 + 1.04i)7-s − 0.353·8-s + 0.333i·9-s + (0.706 + 0.0153i)10-s + 0.826i·11-s + (−0.204 − 0.204i)12-s − 0.581·13-s + (−0.742 − 0.742i)14-s + (0.399 + 0.417i)15-s + 0.250·16-s − 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.995 - 0.0935i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.995 - 0.0935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1003039788\)
\(L(\frac12)\) \(\approx\) \(0.1003039788\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.23 + 0.0484i)T \)
37 \( 1 + (-3.14 + 5.20i)T \)
good7 \( 1 + (-2.77 - 2.77i)T + 7iT^{2} \)
11 \( 1 - 2.74iT - 11T^{2} \)
13 \( 1 + 2.09T + 13T^{2} \)
17 \( 1 + 4.54iT - 17T^{2} \)
19 \( 1 + (3.00 - 3.00i)T - 19iT^{2} \)
23 \( 1 + 1.09T + 23T^{2} \)
29 \( 1 + (3.41 + 3.41i)T + 29iT^{2} \)
31 \( 1 + (0.444 - 0.444i)T - 31iT^{2} \)
41 \( 1 + 3.49iT - 41T^{2} \)
43 \( 1 + 2.64T + 43T^{2} \)
47 \( 1 + (4.28 + 4.28i)T + 47iT^{2} \)
53 \( 1 + (4.10 - 4.10i)T - 53iT^{2} \)
59 \( 1 + (10.1 - 10.1i)T - 59iT^{2} \)
61 \( 1 + (4.77 - 4.77i)T - 61iT^{2} \)
67 \( 1 + (3.42 - 3.42i)T - 67iT^{2} \)
71 \( 1 - 0.512T + 71T^{2} \)
73 \( 1 + (9.57 + 9.57i)T + 73iT^{2} \)
79 \( 1 + (4.84 - 4.84i)T - 79iT^{2} \)
83 \( 1 + (-4.09 + 4.09i)T - 83iT^{2} \)
89 \( 1 + (9.59 + 9.59i)T + 89iT^{2} \)
97 \( 1 + 5.08iT - 97T^{2} \)
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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

−10.29468124653454927206801439239, −9.247814675843724395482420933899, −8.547963291116465815626733714879, −7.60730833733909286096201605163, −7.38113504957230419685437140947, −6.11846620535516535216418045549, −5.13289814621004619543990411481, −4.30395063488006702007135516098, −2.66424224895777219617023441361, −1.69507024233232380349483699140, 0.06086786063625462514964118252, 1.43619314761901943749157815676, 3.21257797818857810395071689573, 4.22265432136236109854221585532, 4.92965928880641262386808064452, 6.25483600060800934507642717847, 7.11886325153072384945614603996, 8.023365390803168424345254920258, 8.367664229579874170311619535469, 9.481599874312167817716777243653

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