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Properties

Label 2-1155-77.76-c1-0-0
Degree $2$
Conductor $1155$
Sign $0.953 + 0.300i$
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  − 2.76i·2-s − i·3-s − 5.66·4-s − i·5-s − 2.76·6-s + (−2.02 − 1.70i)7-s + 10.1i·8-s − 9-s − 2.76·10-s + (−3.06 + 1.27i)11-s + 5.66i·12-s + 3.06·13-s + (−4.72 + 5.59i)14-s − 15-s + 16.7·16-s − 2.89·17-s + ⋯
L(s)  = 1  − 1.95i·2-s − 0.577i·3-s − 2.83·4-s − 0.447i·5-s − 1.13·6-s + (−0.764 − 0.644i)7-s + 3.58i·8-s − 0.333·9-s − 0.875·10-s + (−0.922 + 0.385i)11-s + 1.63i·12-s + 0.850·13-s + (−1.26 + 1.49i)14-s − 0.258·15-s + 4.19·16-s − 0.702·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.005483591965\)
\(L(\frac12)\) \(\approx\) \(0.005483591965\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (2.02 + 1.70i)T \)
11 \( 1 + (3.06 - 1.27i)T \)
good2 \( 1 + 2.76iT - 2T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 + 2.89T + 17T^{2} \)
19 \( 1 + 5.02T + 19T^{2} \)
23 \( 1 - 6.43T + 23T^{2} \)
29 \( 1 + 3.97iT - 29T^{2} \)
31 \( 1 - 5.54iT - 31T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 + 9.03T + 41T^{2} \)
43 \( 1 - 0.633iT - 43T^{2} \)
47 \( 1 + 2.57iT - 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
59 \( 1 - 6.56iT - 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 - 8.52T + 71T^{2} \)
73 \( 1 + 2.75T + 73T^{2} \)
79 \( 1 + 5.94iT - 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 17.6iT - 89T^{2} \)
97 \( 1 - 3.11iT - 97T^{2} \)
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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

−10.15005298553152458648186175165, −8.949688496600377075545301685172, −8.654667378490331621356158940231, −7.52577549218776426842888849105, −6.27623254556547663494332134130, −5.06115432919463670473702812267, −4.25581214156956903797161553571, −3.27343266166904529296898794082, −2.34996102832021115190598964219, −1.20352172163962377419431791501, 0.00268573544950768799393257386, 2.99603908475480514919431309383, 4.01155422641578648143919996282, 5.01625101421108919599084926457, 5.80925258367635867994945467039, 6.43381564314776300045663320148, 7.15082440351004886250628185762, 8.306237118708847554876335354130, 8.724173871521550961292290947361, 9.460308159159106578494883755110

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