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Properties

Label 2-1155-385.384-c1-0-20
Degree $2$
Conductor $1155$
Sign $0.187 - 0.982i$
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  + 0.219·2-s + 3-s − 1.95·4-s + (1.49 − 1.65i)5-s + 0.219·6-s + (−2.26 + 1.36i)7-s − 0.865·8-s + 9-s + (0.328 − 0.363i)10-s + (−3.31 + 0.00596i)11-s − 1.95·12-s + 4.76i·13-s + (−0.495 + 0.300i)14-s + (1.49 − 1.65i)15-s + 3.71·16-s + 1.75i·17-s + ⋯
L(s)  = 1  + 0.154·2-s + 0.577·3-s − 0.976·4-s + (0.670 − 0.742i)5-s + 0.0894·6-s + (−0.855 + 0.517i)7-s − 0.306·8-s + 0.333·9-s + (0.103 − 0.114i)10-s + (−0.999 + 0.00179i)11-s − 0.563·12-s + 1.32i·13-s + (−0.132 + 0.0802i)14-s + (0.386 − 0.428i)15-s + 0.928·16-s + 0.424i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.283584002\)
\(L(\frac12)\) \(\approx\) \(1.283584002\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (-1.49 + 1.65i)T \)
7 \( 1 + (2.26 - 1.36i)T \)
11 \( 1 + (3.31 - 0.00596i)T \)
good2 \( 1 - 0.219T + 2T^{2} \)
13 \( 1 - 4.76iT - 13T^{2} \)
17 \( 1 - 1.75iT - 17T^{2} \)
19 \( 1 - 3.98T + 19T^{2} \)
23 \( 1 - 5.32iT - 23T^{2} \)
29 \( 1 - 7.93iT - 29T^{2} \)
31 \( 1 - 0.416iT - 31T^{2} \)
37 \( 1 - 8.40iT - 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + 4.16iT - 53T^{2} \)
59 \( 1 - 1.47iT - 59T^{2} \)
61 \( 1 + 7.08T + 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 + 5.14T + 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 - 4.99iT - 79T^{2} \)
83 \( 1 + 4.77iT - 83T^{2} \)
89 \( 1 - 8.37iT - 89T^{2} \)
97 \( 1 - 8.91T + 97T^{2} \)
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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

−9.665284752861303195352100863475, −9.180806313623530799958036219163, −8.640848446738237030146599215677, −7.69380534581922870776684305030, −6.53299247584727267885891943503, −5.50802383209483837248126765216, −4.93280483754117165986883926850, −3.81527095219940603198977859630, −2.83472360902798278700883861246, −1.47165537347839076241640398269, 0.51944073890044433302503659986, 2.62644385380660065187209230441, 3.19325958294760487966344819377, 4.26948167105116968572580189535, 5.44269410565219681243569849731, 6.07577758335816327169650306866, 7.35756039294058928957239019724, 7.85232836943997826847107071225, 8.965813733335869264538233321537, 9.697334019221282430050777921095

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