Properties

Label 2-1110-185.68-c1-0-33
Degree $2$
Conductor $1110$
Sign $-0.764 + 0.644i$
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 + (0.914 − 2.04i)5-s + (−0.707 + 0.707i)6-s + (0.867 − 0.867i)7-s − 8-s − 1.00i·9-s + (−0.914 + 2.04i)10-s + 1.09i·11-s + (0.707 − 0.707i)12-s − 4.92·13-s + (−0.867 + 0.867i)14-s + (−0.796 − 2.08i)15-s + 16-s − 2.47i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (0.408 − 0.912i)5-s + (−0.288 + 0.288i)6-s + (0.327 − 0.327i)7-s − 0.353·8-s − 0.333i·9-s + (−0.289 + 0.645i)10-s + 0.331i·11-s + (0.204 − 0.204i)12-s − 1.36·13-s + (−0.231 + 0.231i)14-s + (−0.205 − 0.539i)15-s + 0.250·16-s − 0.600i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045021830\)
\(L(\frac12)\) \(\approx\) \(1.045021830\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.914 + 2.04i)T \)
37 \( 1 + (-0.134 - 6.08i)T \)
good7 \( 1 + (-0.867 + 0.867i)T - 7iT^{2} \)
11 \( 1 - 1.09iT - 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 + 2.47iT - 17T^{2} \)
19 \( 1 + (5.01 + 5.01i)T + 19iT^{2} \)
23 \( 1 + 0.254T + 23T^{2} \)
29 \( 1 + (-4.52 + 4.52i)T - 29iT^{2} \)
31 \( 1 + (-3.38 - 3.38i)T + 31iT^{2} \)
41 \( 1 + 5.96iT - 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-2.51 + 2.51i)T - 47iT^{2} \)
53 \( 1 + (1.28 + 1.28i)T + 53iT^{2} \)
59 \( 1 + (-3.61 - 3.61i)T + 59iT^{2} \)
61 \( 1 + (-0.890 - 0.890i)T + 61iT^{2} \)
67 \( 1 + (8.08 + 8.08i)T + 67iT^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-9.56 + 9.56i)T - 73iT^{2} \)
79 \( 1 + (1.43 + 1.43i)T + 79iT^{2} \)
83 \( 1 + (-7.34 - 7.34i)T + 83iT^{2} \)
89 \( 1 + (9.93 - 9.93i)T - 89iT^{2} \)
97 \( 1 - 5.50iT - 97T^{2} \)
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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

−9.538974153489076563457887706365, −8.619562522584427580645130224848, −8.089155808385489065351416767423, −7.12324758951375057858489356048, −6.48144467939625906280071081685, −5.09264635157636500598251299844, −4.45845649107331615981556007540, −2.73965018392253041756745256164, −1.87917499151925410820078418418, −0.51547348574944345479392448896, 1.90092264835539369013161937533, 2.69078595899261661812810159747, 3.80666432751261673027770978042, 5.09309371189331666745002289604, 6.13133699337804902726941138542, 6.90261336024972671440156679783, 7.932332464421189104146791780649, 8.456293651029094479703793054013, 9.490155766782012977327678331059, 10.14387409094710248044103754539

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