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Properties

Label 2-52-13.2-c2-0-1
Degree $2$
Conductor $52$
Sign $0.648 + 0.760i$
Analytic cond. $1.41689$
Root an. cond. $1.19033$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.982 − 1.70i)3-s + (5.05 − 5.05i)5-s + (0.383 − 0.102i)7-s + (2.56 − 4.44i)9-s + (−3.51 + 13.1i)11-s + (−9.67 + 8.68i)13-s + (−13.5 − 3.63i)15-s + (12.3 + 7.12i)17-s + (0.873 + 3.25i)19-s + (−0.551 − 0.551i)21-s + (−14.3 + 8.28i)23-s − 26.0i·25-s − 27.7·27-s + (22.3 + 38.7i)29-s + (35.2 − 35.2i)31-s + ⋯
L(s)  = 1  + (−0.327 − 0.567i)3-s + (1.01 − 1.01i)5-s + (0.0547 − 0.0146i)7-s + (0.285 − 0.494i)9-s + (−0.319 + 1.19i)11-s + (−0.743 + 0.668i)13-s + (−0.904 − 0.242i)15-s + (0.725 + 0.419i)17-s + (0.0459 + 0.171i)19-s + (−0.0262 − 0.0262i)21-s + (−0.624 + 0.360i)23-s − 1.04i·25-s − 1.02·27-s + (0.772 + 1.33i)29-s + (1.13 − 1.13i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.648 + 0.760i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ 0.648 + 0.760i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06173 - 0.489918i\)
\(L(\frac12)\) \(\approx\) \(1.06173 - 0.489918i\)
\(L(2)\) 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 \( 1 + (9.67 - 8.68i)T \)
good3 \( 1 + (0.982 + 1.70i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-5.05 + 5.05i)T - 25iT^{2} \)
7 \( 1 + (-0.383 + 0.102i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (3.51 - 13.1i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-12.3 - 7.12i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.873 - 3.25i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (14.3 - 8.28i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.3 - 38.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-35.2 + 35.2i)T - 961iT^{2} \)
37 \( 1 + (-9.58 + 35.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (10.8 + 2.91i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (51.5 + 29.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-54.6 - 54.6i)T + 2.20e3iT^{2} \)
53 \( 1 + 63.3T + 2.80e3T^{2} \)
59 \( 1 + (-31.1 + 8.34i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (29.0 - 50.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (100. + 26.8i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (19.7 + 73.8i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (53.5 + 53.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 129.T + 6.24e3T^{2} \)
83 \( 1 + (-48.6 + 48.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (4.08 - 15.2i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (3.73 + 13.9i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−14.98602184093121887593950653482, −13.73659375144869986487195304492, −12.58470571798192213580260154145, −12.11748097438606168077866114249, −10.08088753195169260793194353733, −9.263884912552406607990330105518, −7.54672526709044649767805871760, −6.12631545448794218348224545396, −4.72151037806061079053354842618, −1.68216552996080368050898644314, 2.85025379911861994648802096597, 5.12995329017878747396734941809, 6.36584295003072613351016430950, 8.034139748504862595287452600549, 9.992228061149998248167502452937, 10.36653913913845114020096386314, 11.68042569187926361905982212810, 13.41340728598337660542225511326, 14.16398045713684768582525699296, 15.39798664641048108952316749397

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