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Properties

Label 2-52-52.47-c3-0-6
Degree $2$
Conductor $52$
Sign $-0.00517 - 0.999i$
Analytic cond. $3.06809$
Root an. cond. $1.75159$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.22 + 1.74i)2-s + 3.23i·3-s + (1.93 + 7.76i)4-s + (4.87 + 4.87i)5-s + (−5.62 + 7.20i)6-s + (−22.0 − 22.0i)7-s + (−9.18 + 20.6i)8-s + 16.5·9-s + (2.38 + 19.3i)10-s + (43.5 + 43.5i)11-s + (−25.0 + 6.26i)12-s + (−2.54 − 46.8i)13-s + (−10.7 − 87.6i)14-s + (−15.7 + 15.7i)15-s + (−56.4 + 30.1i)16-s − 39.7i·17-s + ⋯
L(s)  = 1  + (0.788 + 0.615i)2-s + 0.621i·3-s + (0.242 + 0.970i)4-s + (0.435 + 0.435i)5-s + (−0.382 + 0.490i)6-s + (−1.19 − 1.19i)7-s + (−0.405 + 0.913i)8-s + 0.613·9-s + (0.0752 + 0.611i)10-s + (1.19 + 1.19i)11-s + (−0.603 + 0.150i)12-s + (−0.0542 − 0.998i)13-s + (−0.205 − 1.67i)14-s + (−0.271 + 0.271i)15-s + (−0.882 + 0.470i)16-s − 0.567i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00517 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00517 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $-0.00517 - 0.999i$
Analytic conductor: \(3.06809\)
Root analytic conductor: \(1.75159\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :3/2),\ -0.00517 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.44901 + 1.45653i\)
\(L(\frac12)\) \(\approx\) \(1.44901 + 1.45653i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.22 - 1.74i)T \)
13 \( 1 + (2.54 + 46.8i)T \)
good3 \( 1 - 3.23iT - 27T^{2} \)
5 \( 1 + (-4.87 - 4.87i)T + 125iT^{2} \)
7 \( 1 + (22.0 + 22.0i)T + 343iT^{2} \)
11 \( 1 + (-43.5 - 43.5i)T + 1.33e3iT^{2} \)
17 \( 1 + 39.7iT - 4.91e3T^{2} \)
19 \( 1 + (-81.1 + 81.1i)T - 6.85e3iT^{2} \)
23 \( 1 + 99.6T + 1.21e4T^{2} \)
29 \( 1 - 53.5T + 2.43e4T^{2} \)
31 \( 1 + (-1.94 + 1.94i)T - 2.97e4iT^{2} \)
37 \( 1 + (182. - 182. i)T - 5.06e4iT^{2} \)
41 \( 1 + (33.5 + 33.5i)T + 6.89e4iT^{2} \)
43 \( 1 + 339.T + 7.95e4T^{2} \)
47 \( 1 + (-74.4 - 74.4i)T + 1.03e5iT^{2} \)
53 \( 1 + 28.7T + 1.48e5T^{2} \)
59 \( 1 + (-45.2 - 45.2i)T + 2.05e5iT^{2} \)
61 \( 1 + 560.T + 2.26e5T^{2} \)
67 \( 1 + (-332. + 332. i)T - 3.00e5iT^{2} \)
71 \( 1 + (152. - 152. i)T - 3.57e5iT^{2} \)
73 \( 1 + (52.7 - 52.7i)T - 3.89e5iT^{2} \)
79 \( 1 + 544. iT - 4.93e5T^{2} \)
83 \( 1 + (-24.2 + 24.2i)T - 5.71e5iT^{2} \)
89 \( 1 + (-37.1 + 37.1i)T - 7.04e5iT^{2} \)
97 \( 1 + (-1.15e3 - 1.15e3i)T + 9.12e5iT^{2} \)
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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

−15.31627590838554815309997687132, −14.09984363253714910179018865697, −13.25498291324837408347723219787, −12.09341863724920479017498860688, −10.32927105548802630489859556453, −9.546480482066353848628455637735, −7.30262695031169839753786321484, −6.55245156514819001483586094668, −4.64001560116095975289824760432, −3.38394581453390121254749017844, 1.65308697481245385056047127076, 3.62126952943925628733406279699, 5.76794708139868650433248865668, 6.56889560067732688875001858608, 8.975658826702582233248139465226, 9.881187626456736371285000159717, 11.76356594763995985538511060866, 12.35373019222087124206679417665, 13.39211964104906193478178545179, 14.24989952645924177315420170592

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