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Properties

Label 2-1155-5.4-c1-0-16
Degree $2$
Conductor $1155$
Sign $-0.759 - 0.650i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 1.41i·2-s − i·3-s + 0.00241·4-s + (1.69 + 1.45i)5-s + 1.41·6-s + i·7-s + 2.83i·8-s − 9-s + (−2.05 + 2.40i)10-s − 11-s − 0.00241i·12-s + 3.78i·13-s − 1.41·14-s + (1.45 − 1.69i)15-s − 3.99·16-s + 5.30i·17-s + ⋯
L(s)  = 1  + 0.999i·2-s − 0.577i·3-s + 0.00120·4-s + (0.759 + 0.650i)5-s + 0.577·6-s + 0.377i·7-s + 1.00i·8-s − 0.333·9-s + (−0.649 + 0.759i)10-s − 0.301·11-s − 0.000698i·12-s + 1.05i·13-s − 0.377·14-s + (0.375 − 0.438i)15-s − 0.998·16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.781074631\)
\(L(\frac12)\) \(\approx\) \(1.781074631\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-1.69 - 1.45i)T \)
7 \( 1 - iT \)
11 \( 1 + T \)
good2 \( 1 - 1.41iT - 2T^{2} \)
13 \( 1 - 3.78iT - 13T^{2} \)
17 \( 1 - 5.30iT - 17T^{2} \)
19 \( 1 + 4.91T + 19T^{2} \)
23 \( 1 + 8.56iT - 23T^{2} \)
29 \( 1 + 0.303T + 29T^{2} \)
31 \( 1 + 3.53T + 31T^{2} \)
37 \( 1 + 0.951iT - 37T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 - 2.27iT - 43T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 - 9.34iT - 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + 0.569T + 71T^{2} \)
73 \( 1 + 5.14iT - 73T^{2} \)
79 \( 1 + 9.18T + 79T^{2} \)
83 \( 1 + 7.38iT - 83T^{2} \)
89 \( 1 - 0.535T + 89T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−10.18122935850025363983745162211, −8.840523515223704705584637154983, −8.483679912466729989876374218557, −7.38163697311392819737847396680, −6.64981420081343466613826254923, −6.19941276681294847652962476738, −5.47155469713994844816357892461, −4.18671036407527887396504017766, −2.50816082515207830160621980502, −1.97108881949113102280736230502, 0.72776029896198651780708254691, 2.07599316047971949872808763754, 3.05390261423384522986481118119, 4.04186004166282694830007358998, 5.09215354348615542288397108112, 5.83459476397677935655928715615, 6.99992237096941046486741909112, 7.971151299090586170016199643175, 9.075788456202763465626375618407, 9.728345134660164017578649818446

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