Properties

Label 2-52-13.6-c2-0-0
Degree $2$
Conductor $52$
Sign $-0.199 - 0.979i$
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  + (−2.12 + 3.68i)3-s + (−0.923 + 0.923i)5-s + (−2.49 + 9.30i)7-s + (−4.55 − 7.88i)9-s + (13.5 − 3.64i)11-s + (9.39 − 8.98i)13-s + (−1.43 − 5.36i)15-s + (1.66 − 0.962i)17-s + (−21.0 − 5.63i)19-s + (−28.9 − 28.9i)21-s + (38.2 + 22.0i)23-s + 23.2i·25-s + 0.430·27-s + (10.8 − 18.7i)29-s + (6.79 − 6.79i)31-s + ⋯
L(s)  = 1  + (−0.709 + 1.22i)3-s + (−0.184 + 0.184i)5-s + (−0.356 + 1.32i)7-s + (−0.505 − 0.875i)9-s + (1.23 − 0.331i)11-s + (0.722 − 0.690i)13-s + (−0.0958 − 0.357i)15-s + (0.0980 − 0.0566i)17-s + (−1.10 − 0.296i)19-s + (−1.38 − 1.38i)21-s + (1.66 + 0.960i)23-s + 0.931i·25-s + 0.0159·27-s + (0.373 − 0.646i)29-s + (0.219 − 0.219i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $-0.199 - 0.979i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ -0.199 - 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.562213 + 0.688072i\)
\(L(\frac12)\) \(\approx\) \(0.562213 + 0.688072i\)
\(L(2)\) 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 \( 1 + (-9.39 + 8.98i)T \)
good3 \( 1 + (2.12 - 3.68i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.923 - 0.923i)T - 25iT^{2} \)
7 \( 1 + (2.49 - 9.30i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-13.5 + 3.64i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-1.66 + 0.962i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (21.0 + 5.63i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-38.2 - 22.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-10.8 + 18.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.79 + 6.79i)T - 961iT^{2} \)
37 \( 1 + (-8.26 + 2.21i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (1.93 + 7.23i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (49.6 - 28.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (43.2 + 43.2i)T + 2.20e3iT^{2} \)
53 \( 1 - 92.7T + 2.80e3T^{2} \)
59 \( 1 + (-20.7 + 77.5i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (5.30 + 9.18i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (29.6 + 110. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-16.6 - 4.46i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-41.3 - 41.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 10.3T + 6.24e3T^{2} \)
83 \( 1 + (-24.1 + 24.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (91.0 - 24.4i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-22.9 - 6.13i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−15.30330942108762963265637488913, −15.10463252603098451113189074054, −13.15271232771936248871331492053, −11.72004623125682428636354391919, −11.02476804046614056088759870879, −9.619401527308872781222418028155, −8.696579223852185101615272645706, −6.37612840198662026112395594005, −5.22585299635831212712413199166, −3.51878598558349466207242523966, 1.12036708581179630908805969364, 4.21762957514780765396466078110, 6.50218620198839622127120479483, 6.98340279281018184620664244183, 8.660668551243877942474401313737, 10.45304802017805979066343785865, 11.61171630337906866001157961006, 12.63685297771714999624720576768, 13.51807313067159588800674273993, 14.64752010429194816760766510047

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