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Properties

Label 2-52-52.19-c1-0-3
Degree $2$
Conductor $52$
Sign $0.988 + 0.149i$
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  + (1.41 + 0.00757i)2-s + (−1.40 − 0.808i)3-s + (1.99 + 0.0214i)4-s + (−1.52 + 1.52i)5-s + (−1.97 − 1.15i)6-s + (−1.97 − 0.529i)7-s + (2.82 + 0.0454i)8-s + (−0.193 − 0.334i)9-s + (−2.17 + 2.14i)10-s + (1.12 + 4.18i)11-s + (−2.78 − 1.64i)12-s + (−2.92 − 2.10i)13-s + (−2.78 − 0.763i)14-s + (3.37 − 0.904i)15-s + (3.99 + 0.0857i)16-s + (4.14 − 2.39i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.00535i)2-s + (−0.808 − 0.466i)3-s + (0.999 + 0.0107i)4-s + (−0.683 + 0.683i)5-s + (−0.805 − 0.471i)6-s + (−0.746 − 0.199i)7-s + (0.999 + 0.0160i)8-s + (−0.0644 − 0.111i)9-s + (−0.686 + 0.679i)10-s + (0.337 + 1.26i)11-s + (−0.803 − 0.475i)12-s + (−0.811 − 0.584i)13-s + (−0.745 − 0.203i)14-s + (0.871 − 0.233i)15-s + (0.999 + 0.0214i)16-s + (1.00 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.988 + 0.149i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.988 + 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03117 - 0.0775340i\)
\(L(\frac12)\) \(\approx\) \(1.03117 - 0.0775340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.00757i)T \)
13 \( 1 + (2.92 + 2.10i)T \)
good3 \( 1 + (1.40 + 0.808i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.52 - 1.52i)T - 5iT^{2} \)
7 \( 1 + (1.97 + 0.529i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.12 - 4.18i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-4.14 + 2.39i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.603 + 2.25i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.45 + 4.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.94 - 5.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.420 - 0.420i)T + 31iT^{2} \)
37 \( 1 + (1.86 - 0.5i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.401 - 1.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.59 + 9.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.07 - 8.07i)T - 47iT^{2} \)
53 \( 1 + 1.33T + 53T^{2} \)
59 \( 1 + (-6.48 - 1.73i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.358 - 0.620i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.84 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.454 - 1.69i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-5.35 - 5.35i)T + 73iT^{2} \)
79 \( 1 + 1.11iT - 79T^{2} \)
83 \( 1 + (2.45 + 2.45i)T + 83iT^{2} \)
89 \( 1 + (-1.88 + 0.504i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.32 + 0.355i)T + (84.0 + 48.5i)T^{2} \)
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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

−15.18248653915590634523253925795, −14.50887451370504580815831111430, −12.82616423605526563465675958782, −12.23142792106446519220522517668, −11.24463008497347433599224056492, −9.966350835716094971462687731450, −7.29718188652754422938831483400, −6.75904403637835434681483889053, −5.12114502386845759393652525324, −3.24963357269809124529929716008, 3.63876012293311738040768238378, 5.12455908121771661840991352034, 6.21948518456075396653413501307, 8.020089505511492117142404489398, 9.908905248017275690550543377389, 11.36568930905202831303722362697, 11.97116753266679739496474094758, 13.11431564986490854384964579227, 14.38411139733984476871784250514, 15.68829207319060538734518668997

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