Properties

Label 2-52-52.7-c1-0-0
Degree $2$
Conductor $52$
Sign $0.884 - 0.466i$
Analytic cond. $0.415222$
Root an. cond. $0.644377$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (2.36 + 2.36i)5-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (−4.09 − 2.36i)10-s + (−1.59 − 3.23i)13-s + (1.99 − 3.46i)16-s + (−5.13 + 2.96i)17-s + (3 + 3i)18-s + (6.46 + 1.73i)20-s + 6.19i·25-s + (3.36 + 3.83i)26-s + (5.33 − 9.23i)29-s + (−1.46 + 5.46i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (1.05 + 1.05i)5-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (−1.29 − 0.748i)10-s + (−0.443 − 0.896i)13-s + (0.499 − 0.866i)16-s + (−1.24 + 0.718i)17-s + (0.707 + 0.707i)18-s + (1.44 + 0.387i)20-s + 1.23i·25-s + (0.660 + 0.751i)26-s + (0.989 − 1.71i)29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613181 + 0.151676i\)
\(L(\frac12)\) \(\approx\) \(0.613181 + 0.151676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
13 \( 1 + (1.59 + 3.23i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.36 - 2.36i)T + 5iT^{2} \)
7 \( 1 + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.13 - 2.96i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.33 + 9.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (-1.30 - 4.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (11.3 - 3.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 3.53T + 53T^{2} \)
59 \( 1 + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.69 - 13.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.16 + 1.16i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-1.09 - 4.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.83 + 6.83i)T + (-84.0 - 48.5i)T^{2} \)
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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

−15.37585582276120541336245446028, −14.82557050759146208067415828483, −13.53509954359213331310651820790, −11.78873432927377956931661533151, −10.53412890432023641970963521843, −9.769975835584312212024210673018, −8.430056323170363233677259055606, −6.77367751703942809267965650864, −5.93276301048617207890464001909, −2.62947401537451165449889642003, 2.11364378716024042732234045340, 5.09050989394721151313931821163, 6.83866115633538702196975753435, 8.565546882230064109159812458026, 9.277298427706364785580054008499, 10.53082283933156996669328733270, 11.79859551067721359807291982850, 13.04059856680015448780656299975, 14.09412360474624335344821262968, 15.92779806234524622184594174217

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