Properties

Label 2-1155-33.32-c1-0-37
Degree $2$
Conductor $1155$
Sign $0.594 + 0.804i$
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  − 0.261·2-s + (−0.733 − 1.56i)3-s − 1.93·4-s + i·5-s + (0.192 + 0.411i)6-s + i·7-s + 1.03·8-s + (−1.92 + 2.30i)9-s − 0.261i·10-s + (−3.25 + 0.656i)11-s + (1.41 + 3.03i)12-s − 3.73i·13-s − 0.261i·14-s + (1.56 − 0.733i)15-s + 3.59·16-s − 1.46·17-s + ⋯
L(s)  = 1  − 0.185·2-s + (−0.423 − 0.905i)3-s − 0.965·4-s + 0.447i·5-s + (0.0784 + 0.167i)6-s + 0.377i·7-s + 0.364·8-s + (−0.641 + 0.767i)9-s − 0.0828i·10-s + (−0.980 + 0.197i)11-s + (0.408 + 0.874i)12-s − 1.03i·13-s − 0.0700i·14-s + (0.405 − 0.189i)15-s + 0.898·16-s − 0.355·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.594 + 0.804i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7448883555\)
\(L(\frac12)\) \(\approx\) \(0.7448883555\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.733 + 1.56i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (3.25 - 0.656i)T \)
good2 \( 1 + 0.261T + 2T^{2} \)
13 \( 1 + 3.73iT - 13T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 - 5.82iT - 19T^{2} \)
23 \( 1 + 2.98iT - 23T^{2} \)
29 \( 1 - 4.36T + 29T^{2} \)
31 \( 1 - 5.20T + 31T^{2} \)
37 \( 1 - 2.14T + 37T^{2} \)
41 \( 1 - 3.91T + 41T^{2} \)
43 \( 1 - 1.80iT - 43T^{2} \)
47 \( 1 + 8.56iT - 47T^{2} \)
53 \( 1 + 14.0iT - 53T^{2} \)
59 \( 1 + 4.17iT - 59T^{2} \)
61 \( 1 - 0.658iT - 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 0.170iT - 71T^{2} \)
73 \( 1 - 7.57iT - 73T^{2} \)
79 \( 1 + 12.2iT - 79T^{2} \)
83 \( 1 + 5.56T + 83T^{2} \)
89 \( 1 + 7.47iT - 89T^{2} \)
97 \( 1 - 9.21T + 97T^{2} \)
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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

−9.944980420948084098048728281569, −8.381090458789075916955870171389, −8.256060337466597879063588015681, −7.32824498739807133667340436924, −6.23628621599578625732065266246, −5.49881093338496000661912061671, −4.71033859845563401954101513255, −3.27953600126810027626147783089, −2.17424182325188688763964919154, −0.57598065963181983029219518204, 0.802087500802892246848143455099, 2.84245775599550754878592448671, 4.17650172697940532246814300997, 4.62494505043436968925770120281, 5.39208874527061156530267019629, 6.43513253058721651568415611910, 7.62556544634444293453934168526, 8.569595365862186188688960036856, 9.205074569753749499814126751808, 9.766972682571013548530281693823

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