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Properties

Label 2-1155-33.32-c1-0-2
Degree $2$
Conductor $1155$
Sign $-0.441 - 0.897i$
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  − 0.542·2-s + (−1.58 − 0.693i)3-s − 1.70·4-s + i·5-s + (0.861 + 0.376i)6-s − i·7-s + 2.01·8-s + (2.03 + 2.20i)9-s − 0.542i·10-s + (−2.53 − 2.14i)11-s + (2.70 + 1.18i)12-s + 0.789i·13-s + 0.542i·14-s + (0.693 − 1.58i)15-s + 2.31·16-s − 0.309·17-s + ⋯
L(s)  = 1  − 0.383·2-s + (−0.916 − 0.400i)3-s − 0.852·4-s + 0.447i·5-s + (0.351 + 0.153i)6-s − 0.377i·7-s + 0.710·8-s + (0.679 + 0.733i)9-s − 0.171i·10-s + (−0.763 − 0.645i)11-s + (0.781 + 0.341i)12-s + 0.219i·13-s + 0.145i·14-s + (0.179 − 0.409i)15-s + 0.579·16-s − 0.0750·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.441 - 0.897i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2001772462\)
\(L(\frac12)\) \(\approx\) \(0.2001772462\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.58 + 0.693i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (2.53 + 2.14i)T \)
good2 \( 1 + 0.542T + 2T^{2} \)
13 \( 1 - 0.789iT - 13T^{2} \)
17 \( 1 + 0.309T + 17T^{2} \)
19 \( 1 + 4.40iT - 19T^{2} \)
23 \( 1 + 2.63iT - 23T^{2} \)
29 \( 1 - 8.52T + 29T^{2} \)
31 \( 1 + 7.18T + 31T^{2} \)
37 \( 1 + 4.98T + 37T^{2} \)
41 \( 1 - 4.02T + 41T^{2} \)
43 \( 1 - 9.93iT - 43T^{2} \)
47 \( 1 + 2.67iT - 47T^{2} \)
53 \( 1 - 4.44iT - 53T^{2} \)
59 \( 1 - 9.09iT - 59T^{2} \)
61 \( 1 - 7.24iT - 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 3.04iT - 71T^{2} \)
73 \( 1 - 5.23iT - 73T^{2} \)
79 \( 1 - 7.41iT - 79T^{2} \)
83 \( 1 + 3.28T + 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−10.32587712160318695414146145169, −9.252887529138275463680341111717, −8.381084285507624141818937672309, −7.55753621182379638635078105771, −6.82410927283546059308406734433, −5.84197685999878434856459776416, −4.96064539989101058631931293598, −4.18495066348976793554085221158, −2.74582302632775479757651421605, −1.09126488183612804797582708821, 0.14090743617052091655717573315, 1.65104756873099516276441449217, 3.52644229852823752863161859867, 4.53754414929213985831730099054, 5.21610582805841669530754658851, 5.86065425451757972542257040118, 7.13920643251920066729264322090, 8.012232722397705135474360929658, 8.852202085693328028583116358234, 9.629563527065338597450427862652

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