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Properties

Label 2-1155-33.32-c1-0-57
Degree $2$
Conductor $1155$
Sign $0.545 - 0.838i$
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  + 1.88·2-s + (1.57 + 0.728i)3-s + 1.56·4-s + i·5-s + (2.96 + 1.37i)6-s + i·7-s − 0.828·8-s + (1.93 + 2.28i)9-s + 1.88i·10-s + (0.471 − 3.28i)11-s + (2.45 + 1.13i)12-s + 4.81i·13-s + 1.88i·14-s + (−0.728 + 1.57i)15-s − 4.68·16-s + 6.59·17-s + ⋯
L(s)  = 1  + 1.33·2-s + (0.907 + 0.420i)3-s + 0.780·4-s + 0.447i·5-s + (1.21 + 0.561i)6-s + 0.377i·7-s − 0.292·8-s + (0.646 + 0.763i)9-s + 0.596i·10-s + (0.142 − 0.989i)11-s + (0.708 + 0.328i)12-s + 1.33i·13-s + 0.504i·14-s + (−0.188 + 0.405i)15-s − 1.17·16-s + 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.265927174\)
\(L(\frac12)\) \(\approx\) \(4.265927174\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.57 - 0.728i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (-0.471 + 3.28i)T \)
good2 \( 1 - 1.88T + 2T^{2} \)
13 \( 1 - 4.81iT - 13T^{2} \)
17 \( 1 - 6.59T + 17T^{2} \)
19 \( 1 - 3.38iT - 19T^{2} \)
23 \( 1 + 4.14iT - 23T^{2} \)
29 \( 1 - 8.19T + 29T^{2} \)
31 \( 1 + 7.59T + 31T^{2} \)
37 \( 1 + 1.20T + 37T^{2} \)
41 \( 1 - 0.331T + 41T^{2} \)
43 \( 1 + 6.55iT - 43T^{2} \)
47 \( 1 + 7.08iT - 47T^{2} \)
53 \( 1 - 1.29iT - 53T^{2} \)
59 \( 1 + 13.1iT - 59T^{2} \)
61 \( 1 - 1.41iT - 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + 8.50iT - 73T^{2} \)
79 \( 1 + 8.53iT - 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 1.23iT - 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−9.953704263694579771109927859909, −8.996729984487615719220141431364, −8.425467191939277551494786364860, −7.30420481104792763183487899526, −6.34933603962486531016250791883, −5.54228330552804572205887977394, −4.58340807049966169636626309188, −3.63818225047515112969666749841, −3.14487560787202221270049059445, −1.99153202427642548428911759818, 1.22938764401594223146592349022, 2.73886826268844374263180930982, 3.43332670127740080469721271216, 4.39480710304837918472983760126, 5.22723377204554569071165677150, 6.12255554509541558098713912962, 7.26504438589229051482636481368, 7.75641760021868717901586559892, 8.820083956262985462613607812099, 9.638388359883784492818002895392

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