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Properties

Label 2-1155-5.4-c1-0-37
Degree $2$
Conductor $1155$
Sign $-0.999 - 0.0202i$
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  − 2.74i·2-s + i·3-s − 5.54·4-s + (2.23 + 0.0452i)5-s + 2.74·6-s − i·7-s + 9.72i·8-s − 9-s + (0.124 − 6.13i)10-s − 11-s − 5.54i·12-s + 0.748i·13-s − 2.74·14-s + (−0.0452 + 2.23i)15-s + 15.6·16-s − 6.15i·17-s + ⋯
L(s)  = 1  − 1.94i·2-s + 0.577i·3-s − 2.77·4-s + (0.999 + 0.0202i)5-s + 1.12·6-s − 0.377i·7-s + 3.43i·8-s − 0.333·9-s + (0.0393 − 1.94i)10-s − 0.301·11-s − 1.59i·12-s + 0.207i·13-s − 0.733·14-s + (−0.0116 + 0.577i)15-s + 3.90·16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.0202i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.999 - 0.0202i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.312565160\)
\(L(\frac12)\) \(\approx\) \(1.312565160\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.0452i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 + 2.74iT - 2T^{2} \)
13 \( 1 - 0.748iT - 13T^{2} \)
17 \( 1 + 6.15iT - 17T^{2} \)
19 \( 1 - 1.43T + 19T^{2} \)
23 \( 1 + 5.82iT - 23T^{2} \)
29 \( 1 - 0.913T + 29T^{2} \)
31 \( 1 - 2.25T + 31T^{2} \)
37 \( 1 + 3.26iT - 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 + 9.99iT - 43T^{2} \)
47 \( 1 + 8.43iT - 47T^{2} \)
53 \( 1 + 5.64iT - 53T^{2} \)
59 \( 1 - 4.37T + 59T^{2} \)
61 \( 1 + 9.75T + 61T^{2} \)
67 \( 1 - 6.69iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 3.60iT - 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
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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

−9.729086457330241873693788175685, −9.031466021659935783532213804152, −8.325433723242206603248539511710, −6.83047396717967780011505743389, −5.32241534344348374321313461526, −4.92338780654784116772669936528, −3.83452455320125308119095447426, −2.85432149822942360470077646203, −2.07300209693525768040264391268, −0.61611220720048495626815656437, 1.44711347174233575373027868421, 3.27778495565125604818599673449, 4.71268308758379689272478451652, 5.50868632699017895741019495407, 6.15508829275472421658396573480, 6.66775767579979639744838786875, 7.77567226505086494807514632904, 8.224710829737863169647775751507, 9.138884380597367819233206960874, 9.744912533106117806639136496119

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