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Properties

Label 2-1133-1133.505-c0-0-0
Degree $2$
Conductor $1133$
Sign $-0.563 + 0.825i$
Analytic cond. $0.565440$
Root an. cond. $0.751957$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 − 1.56i)3-s + (−0.850 + 0.526i)4-s + (−0.876 + 0.163i)5-s + (−0.784 − 2.75i)9-s + (−0.273 − 0.961i)11-s + (−0.181 + 1.95i)12-s + (−0.781 + 1.56i)15-s + (0.445 − 0.895i)16-s + (0.658 − 0.600i)20-s + (0.538 − 1.89i)23-s + (−0.191 + 0.0741i)25-s + (−3.41 − 1.32i)27-s + (−0.876 + 1.75i)31-s + (−1.83 − 0.710i)33-s + (2.11 + 1.93i)36-s + (0.0170 − 0.183i)37-s + ⋯
L(s)  = 1  + (1.18 − 1.56i)3-s + (−0.850 + 0.526i)4-s + (−0.876 + 0.163i)5-s + (−0.784 − 2.75i)9-s + (−0.273 − 0.961i)11-s + (−0.181 + 1.95i)12-s + (−0.781 + 1.56i)15-s + (0.445 − 0.895i)16-s + (0.658 − 0.600i)20-s + (0.538 − 1.89i)23-s + (−0.191 + 0.0741i)25-s + (−3.41 − 1.32i)27-s + (−0.876 + 1.75i)31-s + (−1.83 − 0.710i)33-s + (2.11 + 1.93i)36-s + (0.0170 − 0.183i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1133\)    =    \(11 \cdot 103\)
Sign: $-0.563 + 0.825i$
Analytic conductor: \(0.565440\)
Root analytic conductor: \(0.751957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1133} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1133,\ (\ :0),\ -0.563 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9487035219\)
\(L(\frac12)\) \(\approx\) \(0.9487035219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.273 + 0.961i)T \)
103 \( 1 + (-0.445 + 0.895i)T \)
good2 \( 1 + (0.850 - 0.526i)T^{2} \)
3 \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \)
5 \( 1 + (0.876 - 0.163i)T + (0.932 - 0.361i)T^{2} \)
7 \( 1 + (-0.0922 + 0.995i)T^{2} \)
13 \( 1 + (-0.0922 + 0.995i)T^{2} \)
17 \( 1 + (-0.739 + 0.673i)T^{2} \)
19 \( 1 + (0.273 - 0.961i)T^{2} \)
23 \( 1 + (-0.538 + 1.89i)T + (-0.850 - 0.526i)T^{2} \)
29 \( 1 + (-0.932 + 0.361i)T^{2} \)
31 \( 1 + (0.876 - 1.75i)T + (-0.602 - 0.798i)T^{2} \)
37 \( 1 + (-0.0170 + 0.183i)T + (-0.982 - 0.183i)T^{2} \)
41 \( 1 + (-0.932 - 0.361i)T^{2} \)
43 \( 1 + (0.982 - 0.183i)T^{2} \)
47 \( 1 - 1.86T + T^{2} \)
53 \( 1 + (-1.02 - 1.35i)T + (-0.273 + 0.961i)T^{2} \)
59 \( 1 + (-0.136 - 0.124i)T + (0.0922 + 0.995i)T^{2} \)
61 \( 1 + (-0.739 + 0.673i)T^{2} \)
67 \( 1 + (-1.09 - 0.995i)T + (0.0922 + 0.995i)T^{2} \)
71 \( 1 + (-1.18 - 0.221i)T + (0.932 + 0.361i)T^{2} \)
73 \( 1 + (-0.932 - 0.361i)T^{2} \)
79 \( 1 + (-0.932 + 0.361i)T^{2} \)
83 \( 1 + (-0.0922 + 0.995i)T^{2} \)
89 \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \)
97 \( 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)T^{2} \)
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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

−9.184287846714661115937015566077, −8.567467404878171469653431340639, −8.274894145717170884905969556985, −7.35998758876557034093630276833, −6.85399051467452409039266544617, −5.60740483975384086590254310539, −4.09115106368821173692068643850, −3.33369192803720840389160023910, −2.53941694450608951368460157893, −0.77557366796882372220143474069, 2.20110880040813928716393870251, 3.66464191131713114156613599938, 4.02437966819712876572613573063, 4.91973080962226871210578706495, 5.56608183928938325699527288315, 7.52311084000322142693913582322, 7.965738106817268308411785014906, 8.945652503808884138811896624072, 9.474390726930071062755580277216, 9.968169570795221416702441161686

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