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Properties

Label 2-1155-5.4-c1-0-32
Degree $2$
Conductor $1155$
Sign $0.994 + 0.0999i$
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  + 1.40i·2-s − i·3-s + 0.0352·4-s + (−2.22 − 0.223i)5-s + 1.40·6-s − i·7-s + 2.85i·8-s − 9-s + (0.313 − 3.11i)10-s + 11-s − 0.0352i·12-s − 1.11i·13-s + 1.40·14-s + (−0.223 + 2.22i)15-s − 3.92·16-s − 3.90i·17-s + ⋯
L(s)  = 1  + 0.991i·2-s − 0.577i·3-s + 0.0176·4-s + (−0.994 − 0.0999i)5-s + 0.572·6-s − 0.377i·7-s + 1.00i·8-s − 0.333·9-s + (0.0990 − 0.986i)10-s + 0.301·11-s − 0.0101i·12-s − 0.309i·13-s + 0.374·14-s + (−0.0577 + 0.574i)15-s − 0.982·16-s − 0.946i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.0999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.0999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.994 + 0.0999i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.994 + 0.0999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.447569427\)
\(L(\frac12)\) \(\approx\) \(1.447569427\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.223i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 - 1.40iT - 2T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + 3.90iT - 17T^{2} \)
19 \( 1 - 6.16T + 19T^{2} \)
23 \( 1 + 5.20iT - 23T^{2} \)
29 \( 1 + 8.42T + 29T^{2} \)
31 \( 1 - 8.95T + 31T^{2} \)
37 \( 1 + 0.540iT - 37T^{2} \)
41 \( 1 - 3.36T + 41T^{2} \)
43 \( 1 + 9.48iT - 43T^{2} \)
47 \( 1 + 3.04iT - 47T^{2} \)
53 \( 1 + 2.86iT - 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 5.35iT - 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 0.333iT - 73T^{2} \)
79 \( 1 + 0.575T + 79T^{2} \)
83 \( 1 - 0.318iT - 83T^{2} \)
89 \( 1 + 6.02T + 89T^{2} \)
97 \( 1 + 0.903iT - 97T^{2} \)
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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

−9.567647097310028361360957787973, −8.521571483557856834256010515415, −7.958370904177123204259329656035, −7.14105125955825810611334858085, −6.85968295004241891963392170945, −5.65020972962621988067955986874, −4.87730469575386157265989544690, −3.64628878634945629892422987921, −2.48203029886106150636261167863, −0.72361933814224443111248512266, 1.23149916713470849924130995468, 2.67884720784436040688637248220, 3.58706018548204600523219398647, 4.16000058767211368220102686992, 5.37763870102371950440825677889, 6.47798867003634695173035255711, 7.44681362247020152893877108755, 8.256229216388457669702206942660, 9.358709111096890780046532269190, 9.816337612584727577427088467050

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