Properties

Label 2-3042-13.12-c1-0-34
Degree $2$
Conductor $3042$
Sign $0.960 - 0.277i$
Analytic cond. $24.2904$
Root an. cond. $4.92853$
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 − 1.73i·5-s − 1.26i·7-s i·8-s + 1.73·10-s + 1.26i·11-s + 1.26·14-s + 16-s + 5.19·17-s + 4.73i·19-s + 1.73i·20-s − 1.26·22-s − 8.19·23-s + 2.00·25-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.774i·5-s − 0.479i·7-s − 0.353i·8-s + 0.547·10-s + 0.382i·11-s + 0.338·14-s + 0.250·16-s + 1.26·17-s + 1.08i·19-s + 0.387i·20-s − 0.270·22-s − 1.70·23-s + 0.400·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3042\)    =    \(2 \cdot 3^{2} \cdot 13^{2}\)
Sign: $0.960 - 0.277i$
Analytic conductor: \(24.2904\)
Root analytic conductor: \(4.92853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3042} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3042,\ (\ :1/2),\ 0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.700817038\)
\(L(\frac12)\) \(\approx\) \(1.700817038\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good5 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + 1.26iT - 7T^{2} \)
11 \( 1 - 1.26iT - 11T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 - 4.73iT - 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 9.46iT - 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 - 4.19T + 43T^{2} \)
47 \( 1 + 4.73iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 7.26iT - 67T^{2} \)
71 \( 1 - 2.19iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 + 5.66iT - 83T^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−8.528197878448980449414463757416, −8.023083020621778224621612373009, −7.32802233318404623955482718754, −6.51413863725197972569277820583, −5.61531728575718966664116971954, −5.07554022201380351734413285253, −4.11283763399458967427802181847, −3.46794654278528226964588061284, −1.88215650738788830666285664808, −0.76342137354852615322798371255, 0.855595296510717856636467497669, 2.27927250007156010545737945339, 2.88026134974295139236782063997, 3.76992010708974854253397926124, 4.66487284234912939850531083849, 5.71791356953175737763660379785, 6.23219618402024318917055737516, 7.29730763584879703496422085345, 8.023147022633971543702964668515, 8.727927020371192527495451909402

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