Properties

Label 2-1155-33.32-c1-0-54
Degree $2$
Conductor $1155$
Sign $0.695 + 0.718i$
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  + 0.371·2-s + (−0.305 + 1.70i)3-s − 1.86·4-s + i·5-s + (−0.113 + 0.634i)6-s i·7-s − 1.43·8-s + (−2.81 − 1.04i)9-s + 0.371i·10-s + (−1.93 + 2.69i)11-s + (0.569 − 3.17i)12-s − 6.33i·13-s − 0.371i·14-s + (−1.70 − 0.305i)15-s + 3.18·16-s + 1.53·17-s + ⋯
L(s)  = 1  + 0.262·2-s + (−0.176 + 0.984i)3-s − 0.930·4-s + 0.447i·5-s + (−0.0464 + 0.258i)6-s − 0.377i·7-s − 0.507·8-s + (−0.937 − 0.347i)9-s + 0.117i·10-s + (−0.583 + 0.811i)11-s + (0.164 − 0.916i)12-s − 1.75i·13-s − 0.0993i·14-s + (−0.440 − 0.0789i)15-s + 0.797·16-s + 0.373·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.695 + 0.718i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7733372699\)
\(L(\frac12)\) \(\approx\) \(0.7733372699\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.305 - 1.70i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (1.93 - 2.69i)T \)
good2 \( 1 - 0.371T + 2T^{2} \)
13 \( 1 + 6.33iT - 13T^{2} \)
17 \( 1 - 1.53T + 17T^{2} \)
19 \( 1 + 1.59iT - 19T^{2} \)
23 \( 1 - 0.203iT - 23T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + 5.91T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 - 9.29T + 41T^{2} \)
43 \( 1 + 9.25iT - 43T^{2} \)
47 \( 1 + 4.56iT - 47T^{2} \)
53 \( 1 - 2.51iT - 53T^{2} \)
59 \( 1 + 4.95iT - 59T^{2} \)
61 \( 1 - 8.69iT - 61T^{2} \)
67 \( 1 + 4.79T + 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 - 7.53iT - 73T^{2} \)
79 \( 1 + 6.74iT - 79T^{2} \)
83 \( 1 - 5.23T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 - 4.65T + 97T^{2} \)
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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

−9.793054548635281052778474720664, −9.094014725009090688265965772990, −8.057211023235772664408591713067, −7.38955354746679628841505519553, −5.87568433045724392752093446556, −5.36710264183682360116237611364, −4.47738144491414388716283232117, −3.62236447088007764080831599451, −2.76175585789339341729468432860, −0.36890569783175310806468322865, 1.17552563011372389659360154472, 2.53144996920219277895313386330, 3.82901121690141610993784574863, 4.84762636804694471561764517577, 5.73866673815668930099061099635, 6.31613897597845335897489560122, 7.59498059129994704495427208144, 8.214742688841337045775259948715, 9.092144225651240195594892141099, 9.519991726541723017005874069436

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