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Properties

Label 2-1155-5.4-c1-0-55
Degree $2$
Conductor $1155$
Sign $0.139 - 0.990i$
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  − 2.21i·2-s − i·3-s − 2.90·4-s + (0.311 − 2.21i)5-s − 2.21·6-s − i·7-s + 2i·8-s − 9-s + (−4.90 − 0.688i)10-s − 11-s + 2.90i·12-s − 1.31i·13-s − 2.21·14-s + (−2.21 − 0.311i)15-s − 1.37·16-s + 1.90i·17-s + ⋯
L(s)  = 1  − 1.56i·2-s − 0.577i·3-s − 1.45·4-s + (0.139 − 0.990i)5-s − 0.903·6-s − 0.377i·7-s + 0.707i·8-s − 0.333·9-s + (−1.55 − 0.217i)10-s − 0.301·11-s + 0.838i·12-s − 0.363i·13-s − 0.591·14-s + (−0.571 − 0.0803i)15-s − 0.344·16-s + 0.461i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.139 - 0.990i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056061793\)
\(L(\frac12)\) \(\approx\) \(1.056061793\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.311 + 2.21i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 + 2.21iT - 2T^{2} \)
13 \( 1 + 1.31iT - 13T^{2} \)
17 \( 1 - 1.90iT - 17T^{2} \)
19 \( 1 + 0.377T + 19T^{2} \)
23 \( 1 + 2.52iT - 23T^{2} \)
29 \( 1 - 6.11T + 29T^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 + 1.31iT - 37T^{2} \)
41 \( 1 + 2.68T + 41T^{2} \)
43 \( 1 + 6.11iT - 43T^{2} \)
47 \( 1 + 1.59iT - 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 7.21T + 59T^{2} \)
61 \( 1 + 1.04T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 4.92T + 71T^{2} \)
73 \( 1 + 12.4iT - 73T^{2} \)
79 \( 1 - 0.969T + 79T^{2} \)
83 \( 1 - 6.71iT - 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 8.54iT - 97T^{2} \)
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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

−9.262994405872423859333527591069, −8.605072509136534513585348520391, −7.78824294238440150530472195498, −6.63797932885554556337189997615, −5.48360041094809188186767334173, −4.55878617272930932342601111859, −3.64716746265621903100955347088, −2.48739692291699389284034118783, −1.50205526758356361744039586439, −0.46147843569848726125009426241, 2.40298280913271964015211146646, 3.58531341388092293939534308781, 4.77015980306479305086737181169, 5.51188350214344190978245405041, 6.35380488338740881512951666108, 6.99046367668587840433272807234, 7.83109085104692184903475522661, 8.613064294093837056404656734154, 9.467260220499005672909203381350, 10.13724888411481620794887155866

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