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Properties

Label 2-1155-7.4-c1-0-26
Degree $2$
Conductor $1155$
Sign $-0.242 - 0.970i$
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  + (0.765 + 1.32i)2-s + (−0.5 + 0.866i)3-s + (−0.171 + 0.297i)4-s + (0.5 + 0.866i)5-s − 1.53·6-s + (1.95 + 1.78i)7-s + 2.53·8-s + (−0.499 − 0.866i)9-s + (−0.765 + 1.32i)10-s + (0.5 − 0.866i)11-s + (−0.171 − 0.297i)12-s + 5.35·13-s + (−0.865 + 3.95i)14-s − 0.999·15-s + (2.28 + 3.95i)16-s + (0.656 − 1.13i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.937i)2-s + (−0.288 + 0.499i)3-s + (−0.0857 + 0.148i)4-s + (0.223 + 0.387i)5-s − 0.624·6-s + (0.739 + 0.673i)7-s + 0.896·8-s + (−0.166 − 0.288i)9-s + (−0.242 + 0.419i)10-s + (0.150 − 0.261i)11-s + (−0.0495 − 0.0857i)12-s + 1.48·13-s + (−0.231 + 1.05i)14-s − 0.258·15-s + (0.571 + 0.989i)16-s + (0.159 − 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.242 - 0.970i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.635856372\)
\(L(\frac12)\) \(\approx\) \(2.635856372\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 1.78i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.765 - 1.32i)T + (-1 + 1.73i)T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 + (-0.656 + 1.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 + 5.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.460 - 0.798i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.16T + 29T^{2} \)
31 \( 1 + (1.75 - 3.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.60 + 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 1.17T + 43T^{2} \)
47 \( 1 + (0.978 + 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.06 - 8.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.32 - 9.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0865 - 0.149i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.55 + 4.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.02T + 71T^{2} \)
73 \( 1 + (-0.192 + 0.333i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.05 + 8.75i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (-0.866 - 1.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.868T + 97T^{2} \)
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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

−10.20626440110160417049456978752, −8.884648036831368715388070775552, −8.493800575143539419367089732443, −7.29900271210130226950274053730, −6.44097811227577473996534908157, −5.86772460733354706150048941771, −5.04507022933425624041215488748, −4.30070915886927000266538349546, −3.03327164766817991473704523007, −1.51270400880296205069306149967, 1.25444130990879436141552405252, 1.85017917305758631402827343505, 3.35985909111341877546741233952, 4.20472431647529938540917316035, 5.01600968659821964217914800631, 6.14073068509793408741560572423, 6.98862875500344968189024963232, 8.191552572863778041877421499776, 8.388298618669615525273831946824, 10.02093960829520766716834899561

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