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Properties

Label 2-1155-5.4-c1-0-7
Degree $2$
Conductor $1155$
Sign $-0.578 + 0.815i$
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  + 2.32i·2-s + i·3-s − 3.42·4-s + (1.29 − 1.82i)5-s − 2.32·6-s + i·7-s − 3.32i·8-s − 9-s + (4.24 + 3.01i)10-s + 11-s − 3.42i·12-s + 0.502i·13-s − 2.32·14-s + (1.82 + 1.29i)15-s + 0.894·16-s + 4.71i·17-s + ⋯
L(s)  = 1  + 1.64i·2-s + 0.577i·3-s − 1.71·4-s + (0.578 − 0.815i)5-s − 0.951·6-s + 0.377i·7-s − 1.17i·8-s − 0.333·9-s + (1.34 + 0.952i)10-s + 0.301·11-s − 0.989i·12-s + 0.139i·13-s − 0.622·14-s + (0.470 + 0.333i)15-s + 0.223·16-s + 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.578 + 0.815i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.578 + 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093345224\)
\(L(\frac12)\) \(\approx\) \(1.093345224\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (-1.29 + 1.82i)T \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 - 2.32iT - 2T^{2} \)
13 \( 1 - 0.502iT - 13T^{2} \)
17 \( 1 - 4.71iT - 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 - 8.45iT - 23T^{2} \)
29 \( 1 + 6.83T + 29T^{2} \)
31 \( 1 + 3.74T + 31T^{2} \)
37 \( 1 - 10.6iT - 37T^{2} \)
41 \( 1 - 4.41T + 41T^{2} \)
43 \( 1 - 2.08iT - 43T^{2} \)
47 \( 1 + 3.12iT - 47T^{2} \)
53 \( 1 + 6.43iT - 53T^{2} \)
59 \( 1 + 9.33T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 - 8.31iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 1.67iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 5.93iT - 83T^{2} \)
89 \( 1 - 4.63T + 89T^{2} \)
97 \( 1 - 6.94iT - 97T^{2} \)
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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

−9.822949651328418769891339903681, −9.280117639582616557553092105807, −8.556803361205621378125422867117, −8.015320797785913824474716694138, −6.87388657057167049863893010540, −5.95362025694958283422587044236, −5.53972059848012466909163580451, −4.61863086102033991022128809591, −3.76648549065850720095059687282, −1.84168029919866533072943538582, 0.45670322277372144816685848527, 1.94619564143001454754687952948, 2.55947723045974012267633879700, 3.59680786187324650560889872205, 4.57034730834741093730645867264, 5.85830079191330379569402600458, 6.79412268544304061463587638465, 7.58143338716190976829891373777, 8.915353523759408531659134089691, 9.387601649037217415811551947481

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