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Properties

Label 2-1155-5.4-c1-0-47
Degree $2$
Conductor $1155$
Sign $-0.486 + 0.873i$
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  − 0.568i·2-s − i·3-s + 1.67·4-s + (1.08 − 1.95i)5-s − 0.568·6-s − i·7-s − 2.08i·8-s − 9-s + (−1.11 − 0.617i)10-s + 11-s − 1.67i·12-s + 4.29i·13-s − 0.568·14-s + (−1.95 − 1.08i)15-s + 2.16·16-s − 6.71i·17-s + ⋯
L(s)  = 1  − 0.401i·2-s − 0.577i·3-s + 0.838·4-s + (0.486 − 0.873i)5-s − 0.231·6-s − 0.377i·7-s − 0.738i·8-s − 0.333·9-s + (−0.351 − 0.195i)10-s + 0.301·11-s − 0.484i·12-s + 1.19i·13-s − 0.151·14-s + (−0.504 − 0.280i)15-s + 0.541·16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.486 + 0.873i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.218321391\)
\(L(\frac12)\) \(\approx\) \(2.218321391\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 + 1.95i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 + 0.568iT - 2T^{2} \)
13 \( 1 - 4.29iT - 13T^{2} \)
17 \( 1 + 6.71iT - 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
23 \( 1 - 1.44iT - 23T^{2} \)
29 \( 1 + 2.91T + 29T^{2} \)
31 \( 1 + 3.49T + 31T^{2} \)
37 \( 1 - 4.94iT - 37T^{2} \)
41 \( 1 - 3.95T + 41T^{2} \)
43 \( 1 + 6.03iT - 43T^{2} \)
47 \( 1 - 2.12iT - 47T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + 2.84T + 59T^{2} \)
61 \( 1 - 0.0957T + 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + 0.0842T + 71T^{2} \)
73 \( 1 - 0.741iT - 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 17.6iT - 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 7.99iT - 97T^{2} \)
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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

−9.430252277071187133978256290022, −8.982848479071886823579608743467, −7.58445989678176309567759905402, −7.16441541722895806520215775844, −6.23254560243337719887097118792, −5.32886016943028709482844510130, −4.23416140321366084611278657956, −2.97378399384850429673827382382, −1.87829645177213361761344122103, −0.991574292081528452643264283516, 1.83737528598388056039304016646, 2.92844104764815109856919535593, 3.70935931400960360267573167752, 5.34570491846891771774385845335, 5.85882273276974287092674912768, 6.59977948761256370699116231512, 7.59352786482212099833615904079, 8.254544367867867286681290364379, 9.368862217749274012949950832904, 10.17394464004353814592297606636

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