Properties

Label 2-1155-33.32-c1-0-34
Degree $2$
Conductor $1155$
Sign $0.803 - 0.594i$
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 + (−1.72 − 0.0861i)3-s + 0.00575·4-s + i·5-s + (−2.44 − 0.121i)6-s i·7-s − 2.82·8-s + (2.98 + 0.297i)9-s + 1.41i·10-s + (2.56 − 2.10i)11-s + (−0.00995 − 0.000495i)12-s + 1.52i·13-s − 1.41i·14-s + (0.0861 − 1.72i)15-s − 4.01·16-s + 0.570·17-s + ⋯
L(s)  = 1  + 1.00·2-s + (−0.998 − 0.0497i)3-s + 0.00287·4-s + 0.447i·5-s + (−1.00 − 0.0497i)6-s − 0.377i·7-s − 0.998·8-s + (0.995 + 0.0992i)9-s + 0.447i·10-s + (0.773 − 0.634i)11-s + (−0.00287 − 0.000143i)12-s + 0.423i·13-s − 0.378i·14-s + (0.0222 − 0.446i)15-s − 1.00·16-s + 0.138·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.803 - 0.594i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.668243976\)
\(L(\frac12)\) \(\approx\) \(1.668243976\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.0861i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (-2.56 + 2.10i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
13 \( 1 - 1.52iT - 13T^{2} \)
17 \( 1 - 0.570T + 17T^{2} \)
19 \( 1 - 1.43iT - 19T^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 - 1.98T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 - 5.70T + 41T^{2} \)
43 \( 1 - 4.31iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 4.89iT - 53T^{2} \)
59 \( 1 + 2.01iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 - 2.03T + 67T^{2} \)
71 \( 1 + 5.02iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + 5.30iT - 79T^{2} \)
83 \( 1 + 1.61T + 83T^{2} \)
89 \( 1 + 17.4iT - 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−9.911178681852261887299228874876, −9.318249742473322221723203590539, −8.065065118037589114379550792164, −7.03192832388717208017171641460, −6.19019977898001460970615781744, −5.76985134542305887406925837086, −4.58838916617165478538532401835, −4.02796259516659247340758478029, −2.95771235164081477653693073182, −1.11678801319544051697906266162, 0.75872847835918746969901804184, 2.51960711386623437604556650539, 3.96161594232710634664025831098, 4.62791641448211169295936244451, 5.26060931612541765004887346518, 6.22167571750897425398469791953, 6.71346958973131634314019692719, 8.052826261680596120585983575529, 9.027014806679463818638118016394, 9.749273493277509074698771758813

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