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Properties

Label 2-1155-33.32-c1-0-5
Degree $2$
Conductor $1155$
Sign $-0.353 - 0.935i$
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  + 1.30·2-s + (−0.259 − 1.71i)3-s − 0.296·4-s + i·5-s + (−0.338 − 2.23i)6-s − i·7-s − 2.99·8-s + (−2.86 + 0.888i)9-s + 1.30i·10-s + (−3.24 + 0.692i)11-s + (0.0769 + 0.507i)12-s + 1.36i·13-s − 1.30i·14-s + (1.71 − 0.259i)15-s − 3.31·16-s + 2.94·17-s + ⋯
L(s)  = 1  + 0.922·2-s + (−0.149 − 0.988i)3-s − 0.148·4-s + 0.447i·5-s + (−0.138 − 0.912i)6-s − 0.377i·7-s − 1.05·8-s + (−0.955 + 0.296i)9-s + 0.412i·10-s + (−0.977 + 0.208i)11-s + (0.0222 + 0.146i)12-s + 0.379i·13-s − 0.348i·14-s + (0.442 − 0.0669i)15-s − 0.829·16-s + 0.714·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.353 - 0.935i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5647288055\)
\(L(\frac12)\) \(\approx\) \(0.5647288055\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.259 + 1.71i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (3.24 - 0.692i)T \)
good2 \( 1 - 1.30T + 2T^{2} \)
13 \( 1 - 1.36iT - 13T^{2} \)
17 \( 1 - 2.94T + 17T^{2} \)
19 \( 1 - 4.15iT - 19T^{2} \)
23 \( 1 - 5.83iT - 23T^{2} \)
29 \( 1 - 2.33T + 29T^{2} \)
31 \( 1 + 7.48T + 31T^{2} \)
37 \( 1 + 7.95T + 37T^{2} \)
41 \( 1 + 6.55T + 41T^{2} \)
43 \( 1 + 4.00iT - 43T^{2} \)
47 \( 1 - 4.08iT - 47T^{2} \)
53 \( 1 - 4.31iT - 53T^{2} \)
59 \( 1 + 2.85iT - 59T^{2} \)
61 \( 1 + 0.425iT - 61T^{2} \)
67 \( 1 - 1.24T + 67T^{2} \)
71 \( 1 - 3.21iT - 71T^{2} \)
73 \( 1 + 7.14iT - 73T^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 + 1.95T + 83T^{2} \)
89 \( 1 - 9.13iT - 89T^{2} \)
97 \( 1 + 0.814T + 97T^{2} \)
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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

−10.18116755535994041095170652880, −9.184472492599379852906880039069, −8.139546900884902503188941850729, −7.45526431442103382313151608591, −6.64037748117499522951137222523, −5.64408543417776244732778008105, −5.18000291339786249063874185638, −3.79763784844834159315647903596, −3.02518803011567953642443663731, −1.72620162820077527933469807974, 0.17620517359610913929192065358, 2.65412098429415067526292160774, 3.45560767221177154632106354539, 4.47407329539321527633891822905, 5.27541596190573017736591619904, 5.54923973764123262453987960588, 6.74620488902021865680381906205, 8.246133075339555051572922528839, 8.716903733255703880136334003900, 9.579387612407191310072104570473

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