Properties

Label 2-1155-33.32-c1-0-68
Degree $2$
Conductor $1155$
Sign $0.621 + 0.783i$
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  + 0.490·2-s + (1.73 − 0.0687i)3-s − 1.75·4-s + i·5-s + (0.849 − 0.0337i)6-s i·7-s − 1.84·8-s + (2.99 − 0.237i)9-s + 0.490i·10-s + (−1.95 − 2.67i)11-s + (−3.04 + 0.120i)12-s − 2.97i·13-s − 0.490i·14-s + (0.0687 + 1.73i)15-s + 2.61·16-s + 3.37·17-s + ⋯
L(s)  = 1  + 0.346·2-s + (0.999 − 0.0396i)3-s − 0.879·4-s + 0.447i·5-s + (0.346 − 0.0137i)6-s − 0.377i·7-s − 0.652·8-s + (0.996 − 0.0792i)9-s + 0.155i·10-s + (−0.590 − 0.807i)11-s + (−0.878 + 0.0348i)12-s − 0.826i·13-s − 0.131i·14-s + (0.0177 + 0.446i)15-s + 0.653·16-s + 0.817·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.621 + 0.783i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.098374990\)
\(L(\frac12)\) \(\approx\) \(2.098374990\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0687i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (1.95 + 2.67i)T \)
good2 \( 1 - 0.490T + 2T^{2} \)
13 \( 1 + 2.97iT - 13T^{2} \)
17 \( 1 - 3.37T + 17T^{2} \)
19 \( 1 + 5.83iT - 19T^{2} \)
23 \( 1 + 0.534iT - 23T^{2} \)
29 \( 1 - 4.74T + 29T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + 8.07T + 41T^{2} \)
43 \( 1 + 2.55iT - 43T^{2} \)
47 \( 1 - 7.04iT - 47T^{2} \)
53 \( 1 + 3.11iT - 53T^{2} \)
59 \( 1 - 5.20iT - 59T^{2} \)
61 \( 1 + 9.92iT - 61T^{2} \)
67 \( 1 - 2.50T + 67T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + 5.53iT - 73T^{2} \)
79 \( 1 + 1.72iT - 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 6.67iT - 89T^{2} \)
97 \( 1 - 3.15T + 97T^{2} \)
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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

−9.797246740418727301576850332378, −8.626802463970849112537187766907, −8.245561963597874129115244244273, −7.36788256609685791923141058246, −6.32151842556460302382460390840, −5.21679624404897493719601368012, −4.37846354388075967700014430855, −3.24836078151633663965095937944, −2.82292220496312520430046529006, −0.812768109456368192866812322320, 1.49529003588283344563367036649, 2.79130129317106757884780102790, 3.85809584680706961208536939776, 4.60666181986961686832241933645, 5.39891363615440570183345751262, 6.56775344587944451776065202230, 7.78475477726469506431511396774, 8.313726367854445194243623984351, 9.027161547528684943602096653542, 9.958339237256782001062701239625

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