Properties

Label 2-1134-21.20-c1-0-20
Degree $2$
Conductor $1134$
Sign $0.861 + 0.506i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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  + i·2-s − 4-s − 1.79·5-s + (2.28 + 1.34i)7-s i·8-s − 1.79i·10-s − 2.40i·11-s − 4.89i·13-s + (−1.34 + 2.28i)14-s + 16-s − 3.66·17-s + 3.01i·19-s + 1.79·20-s + 2.40·22-s − 3.76i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.800·5-s + (0.861 + 0.506i)7-s − 0.353i·8-s − 0.566i·10-s − 0.724i·11-s − 1.35i·13-s + (−0.358 + 0.609i)14-s + 0.250·16-s − 0.888·17-s + 0.692i·19-s + 0.400·20-s + 0.512·22-s − 0.785i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.861 + 0.506i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.861 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.101296400\)
\(L(\frac12)\) \(\approx\) \(1.101296400\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2.28 - 1.34i)T \)
good5 \( 1 + 1.79T + 5T^{2} \)
11 \( 1 + 2.40iT - 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 - 3.01iT - 19T^{2} \)
23 \( 1 + 3.76iT - 23T^{2} \)
29 \( 1 + 6.56iT - 29T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + 8.08T + 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 - 5.13T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 - 0.570T + 67T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + 5.51iT - 97T^{2} \)
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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

−9.595498349684490736095710357107, −8.522377558899195872442923222867, −8.092468521843159191880226313094, −7.55318511959564346381203427047, −6.24178992835011105998742470677, −5.63484384851974207858251660463, −4.60637174410768822904483037425, −3.76874331127379967702387697911, −2.44277573605424480342918516880, −0.52745538572833701771154465593, 1.34023539572199820975066623009, 2.44587862063716822258048139052, 3.95202382473863069819665520428, 4.36392994503983161792084207965, 5.28190643704071286697948150977, 6.87825224159721582314250999627, 7.34453781312353064711171403749, 8.421649231649003695210949385895, 9.064172918572680437524172945975, 9.962214419562297400441608196154

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