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Properties

Label 2-2808-117.25-c1-0-32
Degree $2$
Conductor $2808$
Sign $0.0196 + 0.999i$
Analytic cond. $22.4219$
Root an. cond. $4.73518$
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  + (−1.55 + 0.900i)5-s + (2.30 + 1.32i)7-s + (1.01 + 0.585i)11-s + (−1.18 − 3.40i)13-s − 1.33·17-s − 7.81i·19-s + (−2.83 − 4.91i)23-s + (−0.879 + 1.52i)25-s + (−3.00 + 5.20i)29-s + (2.23 − 1.29i)31-s − 4.78·35-s − 5.90i·37-s + (−6.23 + 3.60i)41-s + (−5.04 + 8.74i)43-s + (−8.20 − 4.73i)47-s + ⋯
L(s)  = 1  + (−0.697 + 0.402i)5-s + (0.869 + 0.502i)7-s + (0.306 + 0.176i)11-s + (−0.328 − 0.944i)13-s − 0.324·17-s − 1.79i·19-s + (−0.591 − 1.02i)23-s + (−0.175 + 0.304i)25-s + (−0.557 + 0.966i)29-s + (0.401 − 0.231i)31-s − 0.808·35-s − 0.970i·37-s + (−0.974 + 0.562i)41-s + (−0.769 + 1.33i)43-s + (−1.19 − 0.691i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0196 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0196 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.0196 + 0.999i$
Analytic conductor: \(22.4219\)
Root analytic conductor: \(4.73518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :1/2),\ 0.0196 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070937413\)
\(L(\frac12)\) \(\approx\) \(1.070937413\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (1.18 + 3.40i)T \)
good5 \( 1 + (1.55 - 0.900i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.30 - 1.32i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.01 - 0.585i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.33T + 17T^{2} \)
19 \( 1 + 7.81iT - 19T^{2} \)
23 \( 1 + (2.83 + 4.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.00 - 5.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.23 + 1.29i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.90iT - 37T^{2} \)
41 \( 1 + (6.23 - 3.60i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.04 - 8.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.20 + 4.73i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + (-10.6 + 6.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.22 + 9.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 + 2.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.35iT - 71T^{2} \)
73 \( 1 + 9.98iT - 73T^{2} \)
79 \( 1 + (2.14 - 3.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.70 + 0.982i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.41iT - 89T^{2} \)
97 \( 1 + (-0.765 - 0.441i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.442142313564456690109976818880, −7.985542733565547900075087463200, −7.07876561507985874634779059621, −6.52629407270167952364738490772, −5.26582412439891512454046288389, −4.84840015426275860941098512402, −3.79775597343031258845908481797, −2.85951461360764670324414178361, −1.93676367258106807212792887080, −0.35743159783519846417673769702, 1.25339230042376802978840835087, 2.13887810016298929943189330347, 3.82522560974117123654221084147, 4.01827435901459167948624833532, 5.01839446966909014441693622308, 5.86137018957191736730379440222, 6.85459861083275654902104733151, 7.57956136817821342160900759558, 8.267211321524379668371132617471, 8.700756736038830509517861936350

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