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Properties

Label 2-52-52.19-c1-0-0
Degree $2$
Conductor $52$
Sign $0.606 - 0.794i$
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  + (−1.22 + 0.713i)2-s + (1.40 + 0.808i)3-s + (0.981 − 1.74i)4-s + (−1.52 + 1.52i)5-s + (−2.28 + 0.0122i)6-s + (1.97 + 0.529i)7-s + (0.0454 + 2.82i)8-s + (−0.193 − 0.334i)9-s + (0.775 − 2.95i)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 + (−2.07 − 3.42i)16-s + (4.14 − 2.39i)17-s + ⋯
L(s)  = 1  + (−0.863 + 0.504i)2-s + (0.808 + 0.466i)3-s + (0.490 − 0.871i)4-s + (−0.683 + 0.683i)5-s + (−0.933 + 0.00500i)6-s + (0.746 + 0.199i)7-s + (0.0160 + 0.999i)8-s + (−0.0644 − 0.111i)9-s + (0.245 − 0.934i)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 + (−0.518 − 0.855i)16-s + (1.00 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.606 - 0.794i$
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: \((2,\ 52,\ (\ :1/2),\ 0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617243 + 0.305248i\)
\(L(\frac12)\) \(\approx\) \(0.617243 + 0.305248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.713i)T \)
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

−15.52669805546921481862509939995, −14.73953795760691380630823724605, −14.09372828780273583470371535050, −11.79767115470111637424356786859, −10.77396044953488470857242216407, −9.538359095407311419350046395038, −8.269409760915520700682352968448, −7.53324229934182138616069448974, −5.58213268231148458854591256083, −3.15574860445914424757682818840, 2.14211040022502057497676337930, 4.40548201337442600628230032669, 7.43126493278058723536061911170, 7.998043916113642991829365457162, 9.156525056005741709596961290536, 10.53898079879101446974573852449, 12.01691159203996794328174945880, 12.66650857567169029401825702843, 14.19016356231526777996065641976, 15.37180616300930110635820461303

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