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Properties

Label 2-1155-385.384-c1-0-27
Degree $2$
Conductor $1155$
Sign $-0.145 - 0.989i$
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  − 2.15·2-s + 3-s + 2.63·4-s + (1.03 + 1.98i)5-s − 2.15·6-s + (−2.36 + 1.18i)7-s − 1.35·8-s + 9-s + (−2.23 − 4.26i)10-s + (2.31 + 2.37i)11-s + 2.63·12-s + 2.85i·13-s + (5.09 − 2.54i)14-s + (1.03 + 1.98i)15-s − 2.33·16-s − 4.53i·17-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.577·3-s + 1.31·4-s + (0.463 + 0.885i)5-s − 0.878·6-s + (−0.894 + 0.447i)7-s − 0.480·8-s + 0.333·9-s + (−0.705 − 1.34i)10-s + (0.697 + 0.716i)11-s + 0.759·12-s + 0.791i·13-s + (1.36 − 0.680i)14-s + (0.267 + 0.511i)15-s − 0.584·16-s − 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.145 - 0.989i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.145 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8990186199\)
\(L(\frac12)\) \(\approx\) \(0.8990186199\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (-1.03 - 1.98i)T \)
7 \( 1 + (2.36 - 1.18i)T \)
11 \( 1 + (-2.31 - 2.37i)T \)
good2 \( 1 + 2.15T + 2T^{2} \)
13 \( 1 - 2.85iT - 13T^{2} \)
17 \( 1 + 4.53iT - 17T^{2} \)
19 \( 1 - 8.19T + 19T^{2} \)
23 \( 1 + 3.73iT - 23T^{2} \)
29 \( 1 - 2.78iT - 29T^{2} \)
31 \( 1 - 8.30iT - 31T^{2} \)
37 \( 1 - 2.54iT - 37T^{2} \)
41 \( 1 - 2.83T + 41T^{2} \)
43 \( 1 + 4.91T + 43T^{2} \)
47 \( 1 + 2.80T + 47T^{2} \)
53 \( 1 + 3.66iT - 53T^{2} \)
59 \( 1 + 2.02iT - 59T^{2} \)
61 \( 1 + 7.23T + 61T^{2} \)
67 \( 1 - 13.7iT - 67T^{2} \)
71 \( 1 + 5.37T + 71T^{2} \)
73 \( 1 - 6.09iT - 73T^{2} \)
79 \( 1 + 9.72iT - 79T^{2} \)
83 \( 1 - 1.96iT - 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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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

−9.882980024010072334838721883406, −9.210497969766076353242786832420, −8.784640722188627007177488553150, −7.43216261258709678537955400669, −7.03531402401505526497997711953, −6.38191288943139505283082373393, −4.92065485060900776789476746030, −3.36316163868417511324068708347, −2.54632367080671463339159336134, −1.42137382250333934815141298540, 0.66299653515889935111811929439, 1.60457392841133138993682287655, 3.07552579109705445676133917937, 4.09579746309914531405646507770, 5.60386867390992951461910460885, 6.40290876957871267248064299900, 7.59651584771076422344891704562, 8.009482980996037623349998410641, 8.957470979171136699823315156184, 9.521497895270433001676818221109

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