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Properties

Label 2-1155-7.4-c1-0-4
Degree $2$
Conductor $1155$
Sign $0.321 + 0.947i$
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  + (1.28 + 2.22i)2-s + (−0.5 + 0.866i)3-s + (−2.28 + 3.95i)4-s + (0.5 + 0.866i)5-s − 2.56·6-s + (−2.09 − 1.61i)7-s − 6.59·8-s + (−0.499 − 0.866i)9-s + (−1.28 + 2.22i)10-s + (0.5 − 0.866i)11-s + (−2.28 − 3.95i)12-s − 1.89·13-s + (0.901 − 6.72i)14-s − 0.999·15-s + (−3.88 − 6.72i)16-s + (−3.57 + 6.18i)17-s + ⋯
L(s)  = 1  + (0.906 + 1.56i)2-s + (−0.288 + 0.499i)3-s + (−1.14 + 1.97i)4-s + (0.223 + 0.387i)5-s − 1.04·6-s + (−0.791 − 0.610i)7-s − 2.33·8-s + (−0.166 − 0.288i)9-s + (−0.405 + 0.702i)10-s + (0.150 − 0.261i)11-s + (−0.659 − 1.14i)12-s − 0.526·13-s + (0.240 − 1.79i)14-s − 0.258·15-s + (−0.970 − 1.68i)16-s + (−0.866 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.321 + 0.947i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.321 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8011877662\)
\(L(\frac12)\) \(\approx\) \(0.8011877662\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.09 + 1.61i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.28 - 2.22i)T + (-1 + 1.73i)T^{2} \)
13 \( 1 + 1.89T + 13T^{2} \)
17 \( 1 + (3.57 - 6.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.89 + 5.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.794 + 1.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.41T + 29T^{2} \)
31 \( 1 + (1.35 - 2.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.24 + 9.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.49T + 41T^{2} \)
43 \( 1 - 7.05T + 43T^{2} \)
47 \( 1 + (-6.81 - 11.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.29 - 7.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.38 - 4.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.68 - 6.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.60 - 2.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.06T + 71T^{2} \)
73 \( 1 + (5.54 - 9.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.31 + 7.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.27T + 83T^{2} \)
89 \( 1 + (-6.80 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−10.56967844880718829388781048464, −9.266931423137605851828413558440, −8.801501367229623762823610002084, −7.52645079555320442404558505153, −6.99061642510667496271117003429, −6.11512090606565831045969837131, −5.71417163371569197867684035116, −4.30267797046286655509377663270, −4.07086474113707050672095074137, −2.77748302776867578516267812788, 0.24914443819646884080214089728, 1.83694011974340893890065112977, 2.54059523654383287400980372211, 3.65153123112254319339523925575, 4.69459432921267473619195912694, 5.48175590155455594446273304455, 6.22827675605482763820919111241, 7.31471813344390005024685416836, 8.734213654242153848431497408600, 9.513008158787437406790612212535

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