Properties

Label 2-1155-33.32-c1-0-91
Degree $2$
Conductor $1155$
Sign $-0.925 + 0.378i$
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  + 1.80·2-s + (−1.28 − 1.15i)3-s + 1.27·4-s i·5-s + (−2.32 − 2.09i)6-s i·7-s − 1.31·8-s + (0.310 + 2.98i)9-s − 1.80i·10-s + (1.44 − 2.98i)11-s + (−1.63 − 1.47i)12-s + 1.34i·13-s − 1.80i·14-s + (−1.15 + 1.28i)15-s − 4.92·16-s − 3.37·17-s + ⋯
L(s)  = 1  + 1.27·2-s + (−0.742 − 0.669i)3-s + 0.635·4-s − 0.447i·5-s + (−0.949 − 0.856i)6-s − 0.377i·7-s − 0.466·8-s + (0.103 + 0.994i)9-s − 0.571i·10-s + (0.434 − 0.900i)11-s + (−0.471 − 0.425i)12-s + 0.374i·13-s − 0.483i·14-s + (−0.299 + 0.332i)15-s − 1.23·16-s − 0.819·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.925 + 0.378i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.374023359\)
\(L(\frac12)\) \(\approx\) \(1.374023359\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 + 1.15i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (-1.44 + 2.98i)T \)
good2 \( 1 - 1.80T + 2T^{2} \)
13 \( 1 - 1.34iT - 13T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
19 \( 1 + 6.55iT - 19T^{2} \)
23 \( 1 + 0.712iT - 23T^{2} \)
29 \( 1 + 0.481T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + 7.74T + 37T^{2} \)
41 \( 1 - 4.14T + 41T^{2} \)
43 \( 1 - 8.37iT - 43T^{2} \)
47 \( 1 + 8.28iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 1.07iT - 59T^{2} \)
61 \( 1 + 9.12iT - 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 3.34iT - 71T^{2} \)
73 \( 1 + 8.92iT - 73T^{2} \)
79 \( 1 + 17.6iT - 79T^{2} \)
83 \( 1 + 0.931T + 83T^{2} \)
89 \( 1 + 5.94iT - 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−9.237362456044071675043722763028, −8.699026970256819413192208172516, −7.38991561115147940470017207499, −6.65043923954053594908036435301, −5.95875268771688824005857726390, −5.07513928364982328806610661881, −4.44274826515659498934582872420, −3.37004055604848739355732670701, −2.02215861544991435267524983509, −0.40624692755254658250750613534, 2.10121340265966156612166539895, 3.55640060490460944594543345901, 4.00790113621140968494147860796, 5.10379467082046086530998406428, 5.65790546494824509471370714589, 6.48791566333818493322753591986, 7.25792380964826204980863508587, 8.703698056145094552456194353203, 9.503154648368264231889615827121, 10.28550207698934318147987610236

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