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Properties

Label 2-1155-5.4-c1-0-3
Degree $2$
Conductor $1155$
Sign $-0.662 + 0.749i$
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  + 1.67i·2-s − i·3-s − 0.806·4-s + (−1.48 + 1.67i)5-s + 1.67·6-s − i·7-s + 1.99i·8-s − 9-s + (−2.80 − 2.48i)10-s − 11-s + 0.806i·12-s + 0.481i·13-s + 1.67·14-s + (1.67 + 1.48i)15-s − 4.96·16-s − 0.193i·17-s + ⋯
L(s)  = 1  + 1.18i·2-s − 0.577i·3-s − 0.403·4-s + (−0.662 + 0.749i)5-s + 0.683·6-s − 0.377i·7-s + 0.707i·8-s − 0.333·9-s + (−0.887 − 0.784i)10-s − 0.301·11-s + 0.232i·12-s + 0.133i·13-s + 0.447·14-s + (0.432 + 0.382i)15-s − 1.24·16-s − 0.0470i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.662 + 0.749i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4142987651\)
\(L(\frac12)\) \(\approx\) \(0.4142987651\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 1.67i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 - 1.67iT - 2T^{2} \)
13 \( 1 - 0.481iT - 13T^{2} \)
17 \( 1 + 0.193iT - 17T^{2} \)
19 \( 1 + 3.96T + 19T^{2} \)
23 \( 1 - 3.15iT - 23T^{2} \)
29 \( 1 - 0.130T + 29T^{2} \)
31 \( 1 + 8.24T + 31T^{2} \)
37 \( 1 - 0.481iT - 37T^{2} \)
41 \( 1 + 4.48T + 41T^{2} \)
43 \( 1 + 0.130iT - 43T^{2} \)
47 \( 1 + 1.28iT - 47T^{2} \)
53 \( 1 - 3.64iT - 53T^{2} \)
59 \( 1 - 3.32T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 + 4.64iT - 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 - 5.23iT - 83T^{2} \)
89 \( 1 + 1.79T + 89T^{2} \)
97 \( 1 + 5.21iT - 97T^{2} \)
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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

−10.46568660458584958866207922119, −9.161469056534051674703283136958, −8.270773626705571056916073045727, −7.61266270967110534216834560433, −7.07586092567107774747147430453, −6.39713082125315894141176182095, −5.55714947963450922774387775673, −4.42777835885310003762430264438, −3.24999620053839923833270164598, −2.01564658542379749353381285674, 0.16715875423941808764067629880, 1.76017486186065429908705472011, 2.93007925624338390692724037891, 3.87371687907968260713520300786, 4.60088196102274483627507142139, 5.58461677053885334330195676024, 6.77704656866288443563220813959, 7.901413298930742672904511962140, 8.778012518258437028293145163330, 9.342217995210256395312754265029

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