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Properties

Label 2-52-13.9-c1-0-1
Degree $2$
Conductor $52$
Sign $0.872 + 0.488i$
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 − 1.73i)3-s − 3·5-s + (2 + 3.46i)7-s + (−0.499 − 0.866i)9-s + (−3.5 − 0.866i)13-s + (−3 + 5.19i)15-s + (−1.5 − 2.59i)17-s + (−1 − 1.73i)19-s + 7.99·21-s + (3 − 5.19i)23-s + 4·25-s + 4.00·27-s + (−4.5 + 7.79i)29-s + 2·31-s + (−6 − 10.3i)35-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s − 1.34·5-s + (0.755 + 1.30i)7-s + (−0.166 − 0.288i)9-s + (−0.970 − 0.240i)13-s + (−0.774 + 1.34i)15-s + (−0.363 − 0.630i)17-s + (−0.229 − 0.397i)19-s + 1.74·21-s + (0.625 − 1.08i)23-s + 0.800·25-s + 0.769·27-s + (−0.835 + 1.44i)29-s + 0.359·31-s + (−1.01 − 1.75i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.836004 - 0.218271i\)
\(L(\frac12)\) \(\approx\) \(0.836004 - 0.218271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 0.866i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.11801567263195803439363895937, −14.51304520322972450289738422448, −12.86752542133299052893031973778, −12.13014409463477209152161257592, −11.13283845564273967027794116457, −8.932358985977772946319788991758, −8.028218456497603633589163853291, −7.06391108555236735802437630970, −4.89815905049700458481168744829, −2.57495322641041371625158816099, 3.76958376181139952163545686543, 4.54350086596736237636835442352, 7.29648802658163661817734164938, 8.230182993785509707012128489470, 9.748122364575163662300444350839, 10.86231622955014444559054444162, 11.89317232326513909860766786338, 13.56290443933123066594312249450, 14.86969352570688322322145549390, 15.26200544484071542123120556530

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