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Properties

Label 2-52e2-13.12-c1-0-69
Degree $2$
Conductor $2704$
Sign $-0.832 - 0.554i$
Analytic cond. $21.5915$
Root an. cond. $4.64667$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − 4i·7-s − 3·9-s − 4i·11-s − 3·17-s − 4·23-s + 4·25-s − 29-s + 4i·31-s + 4·35-s + 3i·37-s + 9i·41-s − 8·43-s − 3i·45-s + 8i·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.51i·7-s − 9-s − 1.20i·11-s − 0.727·17-s − 0.834·23-s + 0.800·25-s − 0.185·29-s + 0.718i·31-s + 0.676·35-s + 0.493i·37-s + 1.40i·41-s − 1.21·43-s − 0.447i·45-s + 1.16i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2704\)    =    \(2^{4} \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(21.5915\)
Root analytic conductor: \(4.64667\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2704} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2704,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 9iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.286889706379832922663640540182, −7.74324969950954866391423347703, −6.67025700886463058125137020993, −6.34327232334369413726302767679, −5.26297422473096205932181692944, −4.33328282280001880555721790039, −3.42345358205253223299657707854, −2.79695592966418067312060783839, −1.24757403827324554137466200117, 0, 1.96436314066399732622078199988, 2.46951662513606843658717809492, 3.67480412635643553567613405086, 4.81823495389662608287982543811, 5.35572706140883841840667249209, 6.13204976124694715392804274147, 6.92578603339205590747498603287, 8.001072712767556716062830166160, 8.610769824295248471041951273504

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