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Properties

Label 2-1155-7.2-c1-0-45
Degree $2$
Conductor $1155$
Sign $-0.924 + 0.381i$
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  + (0.967 − 1.67i)2-s + (−0.5 − 0.866i)3-s + (−0.871 − 1.50i)4-s + (0.5 − 0.866i)5-s − 1.93·6-s + (2.59 + 0.512i)7-s + 0.497·8-s + (−0.499 + 0.866i)9-s + (−0.967 − 1.67i)10-s + (0.5 + 0.866i)11-s + (−0.871 + 1.50i)12-s − 6.71·13-s + (3.36 − 3.85i)14-s − 0.999·15-s + (2.22 − 3.85i)16-s + (−0.742 − 1.28i)17-s + ⋯
L(s)  = 1  + (0.683 − 1.18i)2-s + (−0.288 − 0.499i)3-s + (−0.435 − 0.754i)4-s + (0.223 − 0.387i)5-s − 0.789·6-s + (0.981 + 0.193i)7-s + 0.176·8-s + (−0.166 + 0.288i)9-s + (−0.305 − 0.529i)10-s + (0.150 + 0.261i)11-s + (−0.251 + 0.435i)12-s − 1.86·13-s + (0.900 − 1.02i)14-s − 0.258·15-s + (0.556 − 0.963i)16-s + (−0.180 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.924 + 0.381i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.924 + 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.347839394\)
\(L(\frac12)\) \(\approx\) \(2.347839394\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 0.512i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.967 + 1.67i)T + (-1 - 1.73i)T^{2} \)
13 \( 1 + 6.71T + 13T^{2} \)
17 \( 1 + (0.742 + 1.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.15 + 7.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.04 + 3.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.69T + 29T^{2} \)
31 \( 1 + (1.10 + 1.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.10 + 3.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.16T + 41T^{2} \)
43 \( 1 - 4.93T + 43T^{2} \)
47 \( 1 + (-1.92 + 3.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.11 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0703 + 0.121i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.38 - 9.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.23 - 9.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (-0.880 - 1.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.28 + 5.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.32T + 83T^{2} \)
89 \( 1 + (1.59 - 2.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−9.579455316867620867067293524400, −8.867078730122121916462408413150, −7.42709921841979509344658033452, −7.27718624919111461620991106917, −5.58102074313636452857749482783, −4.89731826054458164588101941656, −4.38427153242797263208293651793, −2.70331813495086210751270162501, −2.18680672422598374509462398284, −0.876688318905252980800585737357, 1.76478294113836293745568700057, 3.41888497648640234214345468400, 4.43130990505422231577423014854, 5.23930023011813856965246321844, 5.70066492347703788773985484243, 6.79859912742395823666617813080, 7.56054738310009057119965116101, 8.075765425253619917812243763530, 9.382288663607734864206768905520, 10.13255103818150050300722872657

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