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Properties

Label 2-1155-33.32-c1-0-21
Degree $2$
Conductor $1155$
Sign $0.740 - 0.671i$
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  − 1.41·2-s + (−1.72 + 0.0861i)3-s + 0.00575·4-s − i·5-s + (2.44 − 0.121i)6-s − i·7-s + 2.82·8-s + (2.98 − 0.297i)9-s + 1.41i·10-s + (−2.56 + 2.10i)11-s + (−0.00995 + 0.000495i)12-s + 1.52i·13-s + 1.41i·14-s + (0.0861 + 1.72i)15-s − 4.01·16-s − 0.570·17-s + ⋯
L(s)  = 1  − 1.00·2-s + (−0.998 + 0.0497i)3-s + 0.00287·4-s − 0.447i·5-s + (1.00 − 0.0497i)6-s − 0.377i·7-s + 0.998·8-s + (0.995 − 0.0992i)9-s + 0.447i·10-s + (−0.773 + 0.634i)11-s + (−0.00287 + 0.000143i)12-s + 0.423i·13-s + 0.378i·14-s + (0.0222 + 0.446i)15-s − 1.00·16-s − 0.138·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4051656402\)
\(L(\frac12)\) \(\approx\) \(0.4051656402\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.0861i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (2.56 - 2.10i)T \)
good2 \( 1 + 1.41T + 2T^{2} \)
13 \( 1 - 1.52iT - 13T^{2} \)
17 \( 1 + 0.570T + 17T^{2} \)
19 \( 1 - 1.43iT - 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 7.97T + 29T^{2} \)
31 \( 1 - 1.98T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 - 4.31iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 - 4.89iT - 53T^{2} \)
59 \( 1 - 2.01iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 - 2.03T + 67T^{2} \)
71 \( 1 - 5.02iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + 5.30iT - 79T^{2} \)
83 \( 1 - 1.61T + 83T^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−9.984906210444633931048510488495, −9.213575584159347639107399807129, −8.274219593115718582290284039058, −7.49900567296705207205316048336, −6.77284838098811288497856058532, −5.63890987818653409557447481048, −4.68967646450527501580245046232, −4.12789019719954580206254658860, −2.06472326525209436575232175714, −0.802842699070669242877930554324, 0.43156020898374166631270834251, 1.88525182925861737486919878647, 3.42058520852926685454965550702, 4.73717297793516511311694150594, 5.54400803495724850209577053866, 6.36101740787202338350376633891, 7.55267465173298194749157234166, 7.85111785944262730945772755257, 9.069480739691312842088732164570, 9.727851363656421059543233706362

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