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Properties

Label 2-1155-1155.1154-c0-0-10
Degree $2$
Conductor $1155$
Sign $i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + 4-s − i·5-s − i·7-s − 9-s + 11-s − i·12-s + 2i·13-s − 15-s + 16-s − i·20-s − 21-s − 25-s + i·27-s − i·28-s − 2·29-s + ⋯
L(s)  = 1  − i·3-s + 4-s − i·5-s − i·7-s − 9-s + 11-s − i·12-s + 2i·13-s − 15-s + 16-s − i·20-s − 21-s − 25-s + i·27-s − i·28-s − 2·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308944822\)
\(L(\frac12)\) \(\approx\) \(1.308944822\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 - T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.527384056467353278988998414419, −9.028362675597971356073101626326, −7.894339687283806698440031033957, −7.25569971866949852766409564524, −6.58318811242114638801919333938, −5.88377385981323561375903562525, −4.51667698211988893014933190663, −3.60117938565520584711105417193, −1.94184654146782798660644617949, −1.35306713622553753215455472887, 2.19347572953911401320311710685, 3.11752923669926194729028171336, 3.72476401610692384551673508445, 5.47183495480473603198169839811, 5.78551983750898397389696896877, 6.78069025635358036221574517904, 7.75499352573368707785935945192, 8.581791916309551491610839674765, 9.625876145539839012583007961089, 10.23467578696084954665378544818

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