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Properties

Label 2-1155-5.4-c1-0-20
Degree $2$
Conductor $1155$
Sign $-0.994 - 0.103i$
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  + 1.76i·2-s + i·3-s − 1.09·4-s + (2.22 + 0.230i)5-s − 1.76·6-s − i·7-s + 1.58i·8-s − 9-s + (−0.405 + 3.91i)10-s − 11-s − 1.09i·12-s + 4.21i·13-s + 1.76·14-s + (−0.230 + 2.22i)15-s − 4.99·16-s + 1.43i·17-s + ⋯
L(s)  = 1  + 1.24i·2-s + 0.577i·3-s − 0.548·4-s + (0.994 + 0.103i)5-s − 0.718·6-s − 0.377i·7-s + 0.561i·8-s − 0.333·9-s + (−0.128 + 1.23i)10-s − 0.301·11-s − 0.316i·12-s + 1.16i·13-s + 0.470·14-s + (−0.0595 + 0.574i)15-s − 1.24·16-s + 0.347i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.994 - 0.103i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.994 - 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.896347256\)
\(L(\frac12)\) \(\approx\) \(1.896347256\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.230i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 - 1.76iT - 2T^{2} \)
13 \( 1 - 4.21iT - 13T^{2} \)
17 \( 1 - 1.43iT - 17T^{2} \)
19 \( 1 - 6.78T + 19T^{2} \)
23 \( 1 + 2.64iT - 23T^{2} \)
29 \( 1 + 8.97T + 29T^{2} \)
31 \( 1 + 0.288T + 31T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 - 1.62T + 41T^{2} \)
43 \( 1 - 8.21iT - 43T^{2} \)
47 \( 1 - 3.94iT - 47T^{2} \)
53 \( 1 + 5.94iT - 53T^{2} \)
59 \( 1 + 6.45T + 59T^{2} \)
61 \( 1 - 4.07T + 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 + 8.54iT - 73T^{2} \)
79 \( 1 - 0.690T + 79T^{2} \)
83 \( 1 - 8.67iT - 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 15.4iT - 97T^{2} \)
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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

−9.779693359450130290523962340815, −9.450366927507146638574018291165, −8.470314351245572565625619109929, −7.59042075705302410991330467728, −6.75313361745431900623534605922, −6.07704958079870235587960640771, −5.24557620001733690123774533647, −4.52958998156919131669851304954, −3.13327573180963163316580726797, −1.80098394770434901621786005236, 0.823846758077685001885359055378, 1.96627760220310997605812511589, 2.75293191795838994860856354941, 3.66073750117292155547791382783, 5.36726227176918692939224923016, 5.66953247245520336812361036919, 7.01332796554752932673659511753, 7.70034487465221292460905715802, 8.991263802241034270070597844923, 9.521972544060052630537332172071

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