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Properties

Label 2-1155-33.32-c1-0-85
Degree $2$
Conductor $1155$
Sign $-0.174 + 0.984i$
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 − 1.41i)3-s + i·5-s + (1.41 − 2.00i)6-s − i·7-s − 2.82·8-s + (−1.00 − 2.82i)9-s + 1.41i·10-s + (3 − 1.41i)11-s − 6.24i·13-s − 1.41i·14-s + (1.41 + i)15-s − 4.00·16-s + 0.171·17-s + (−1.41 − 4.00i)18-s + 5.24i·19-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.577 − 0.816i)3-s + 0.447i·5-s + (0.577 − 0.816i)6-s − 0.377i·7-s − 0.999·8-s + (−0.333 − 0.942i)9-s + 0.447i·10-s + (0.904 − 0.426i)11-s − 1.73i·13-s − 0.377i·14-s + (0.365 + 0.258i)15-s − 1.00·16-s + 0.0416·17-s + (−0.333 − 0.942i)18-s + 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.174 + 0.984i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.632932893\)
\(L(\frac12)\) \(\approx\) \(2.632932893\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (-3 + 1.41i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
13 \( 1 + 6.24iT - 13T^{2} \)
17 \( 1 - 0.171T + 17T^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + 7.24iT - 23T^{2} \)
29 \( 1 + 0.171T + 29T^{2} \)
31 \( 1 + 0.242T + 31T^{2} \)
37 \( 1 + 0.242T + 37T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 - 1.58iT - 53T^{2} \)
59 \( 1 + 6.89iT - 59T^{2} \)
61 \( 1 - 0.757iT - 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 2.24iT - 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−9.539721305503979300424156208157, −8.478668509653249009955671111495, −7.998364212252007103274881862779, −6.85548201055275724077570855081, −6.17183920029532488165661404975, −5.42867850452890902026133363272, −4.02904239776062729652291232917, −3.41231515466987706789409162710, −2.50864091686991139863956919717, −0.78793542179436638270893566318, 1.93158175080711857694150068957, 3.16103206689320127691034325136, 4.17965072444425349896857681491, 4.55585656227159295387887675320, 5.47673866272771931103954589533, 6.45062569084525230487493477393, 7.51057914493737260782541589204, 8.803852865376122278531821909257, 9.285382187387869096854291943883, 9.532481851879196631348775001604

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