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Properties

Label 2-1155-77.76-c1-0-61
Degree $2$
Conductor $1155$
Sign $0.0808 - 0.996i$
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  − 2.40i·2-s − i·3-s − 3.78·4-s − i·5-s − 2.40·6-s + (1.69 − 2.03i)7-s + 4.29i·8-s − 9-s − 2.40·10-s + (2.71 − 1.90i)11-s + 3.78i·12-s − 4.65·13-s + (−4.89 − 4.06i)14-s − 15-s + 2.76·16-s + 0.659·17-s + ⋯
L(s)  = 1  − 1.70i·2-s − 0.577i·3-s − 1.89·4-s − 0.447i·5-s − 0.982·6-s + (0.638 − 0.769i)7-s + 1.51i·8-s − 0.333·9-s − 0.760·10-s + (0.818 − 0.574i)11-s + 1.09i·12-s − 1.28·13-s + (−1.30 − 1.08i)14-s − 0.258·15-s + 0.691·16-s + 0.159·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0808 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0808 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0808 - 0.996i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.0808 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106330548\)
\(L(\frac12)\) \(\approx\) \(1.106330548\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-1.69 + 2.03i)T \)
11 \( 1 + (-2.71 + 1.90i)T \)
good2 \( 1 + 2.40iT - 2T^{2} \)
13 \( 1 + 4.65T + 13T^{2} \)
17 \( 1 - 0.659T + 17T^{2} \)
19 \( 1 + 0.0804T + 19T^{2} \)
23 \( 1 + 3.80T + 23T^{2} \)
29 \( 1 + 8.10iT - 29T^{2} \)
31 \( 1 - 3.95iT - 31T^{2} \)
37 \( 1 + 5.99T + 37T^{2} \)
41 \( 1 - 4.39T + 41T^{2} \)
43 \( 1 + 5.26iT - 43T^{2} \)
47 \( 1 - 7.38iT - 47T^{2} \)
53 \( 1 + 3.67T + 53T^{2} \)
59 \( 1 - 6.62iT - 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 6.71T + 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 + 2.35T + 83T^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 + 4.04iT - 97T^{2} \)
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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

−9.452166910137395127104544758013, −8.532128578684879966439390402936, −7.77647274223229261647258202445, −6.75232425853565382702488543471, −5.42889613337980111352342375665, −4.44767384833218167064869497092, −3.73385104395590165161087689026, −2.47944719164780661960575761231, −1.52818020157613534646134899946, −0.49177510174663282390199877813, 2.22830548362252470497454786393, 3.79755661393953907963615395756, 4.83748622769062573067576636867, 5.32675484896999339918930416047, 6.32200194259074633300793849983, 7.07567832809181452384137137293, 7.83947452788554352097439054807, 8.627374292474857476332151724201, 9.405546823080520969196853672717, 9.972492154300159014324157441817

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