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Properties

Label 2-1155-7.4-c1-0-15
Degree $2$
Conductor $1155$
Sign $0.0264 - 0.999i$
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  + (0.448 + 0.776i)2-s + (−0.5 + 0.866i)3-s + (0.598 − 1.03i)4-s + (−0.5 − 0.866i)5-s − 0.896·6-s + (1.40 + 2.24i)7-s + 2.86·8-s + (−0.499 − 0.866i)9-s + (0.448 − 0.776i)10-s + (−0.5 + 0.866i)11-s + (0.598 + 1.03i)12-s − 3.67·13-s + (−1.10 + 2.09i)14-s + 0.999·15-s + (0.0874 + 0.151i)16-s + (−2.55 + 4.42i)17-s + ⋯
L(s)  = 1  + (0.316 + 0.548i)2-s + (−0.288 + 0.499i)3-s + (0.299 − 0.518i)4-s + (−0.223 − 0.387i)5-s − 0.365·6-s + (0.531 + 0.846i)7-s + 1.01·8-s + (−0.166 − 0.288i)9-s + (0.141 − 0.245i)10-s + (−0.150 + 0.261i)11-s + (0.172 + 0.299i)12-s − 1.02·13-s + (−0.296 + 0.560i)14-s + 0.258·15-s + (0.0218 + 0.0378i)16-s + (−0.620 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0264 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0264 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0264 - 0.999i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.0264 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.881338808\)
\(L(\frac12)\) \(\approx\) \(1.881338808\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 2.24i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.448 - 0.776i)T + (-1 + 1.73i)T^{2} \)
13 \( 1 + 3.67T + 13T^{2} \)
17 \( 1 + (2.55 - 4.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.36 - 2.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.19 - 7.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.89T + 29T^{2} \)
31 \( 1 + (-4.73 + 8.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.11 - 8.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.56T + 41T^{2} \)
43 \( 1 + 0.827T + 43T^{2} \)
47 \( 1 + (2.75 + 4.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.46 - 7.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.93 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.21 + 5.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.425 - 0.737i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.38T + 71T^{2} \)
73 \( 1 + (-0.628 + 1.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.75 + 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.93T + 83T^{2} \)
89 \( 1 + (-3.28 - 5.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.98T + 97T^{2} \)
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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

−9.939597666637497598199761216866, −9.302295581880607694216066881423, −8.186414580885749672422445725161, −7.56072522390188421809959403472, −6.38360707838214232870530693058, −5.74092382866345560519156754280, −4.86311035713782188568158917330, −4.40057272876791912874675865933, −2.73491427120828215122176827274, −1.44638876143148860711824838572, 0.819695729421571813834538856789, 2.42949317864287380208662053777, 3.03592249396625867424614972935, 4.53544989036152916595807809428, 4.85455278339234103692827923747, 6.63069430060836356251948600520, 7.03701065450131777144743419595, 7.78369392336046895054880623008, 8.572814625441455980698498752953, 9.856858786706376101307578429653

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