Properties

Label 2-52-52.19-c1-0-2
Degree $2$
Conductor $52$
Sign $0.873 + 0.487i$
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  + (−0.785 − 1.17i)2-s + (1.81 + 1.04i)3-s + (−0.766 + 1.84i)4-s + (0.894 − 0.894i)5-s + (−0.193 − 2.95i)6-s + (−4.37 − 1.17i)7-s + (2.77 − 0.549i)8-s + (0.693 + 1.20i)9-s + (−1.75 − 0.349i)10-s + (0.404 + 1.51i)11-s + (−3.32 + 2.54i)12-s + (−2.03 + 2.97i)13-s + (2.05 + 6.06i)14-s + (2.55 − 0.685i)15-s + (−2.82 − 2.83i)16-s + (−0.0484 + 0.0279i)17-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (1.04 + 0.604i)3-s + (−0.383 + 0.923i)4-s + (0.399 − 0.399i)5-s + (−0.0788 − 1.20i)6-s + (−1.65 − 0.442i)7-s + (0.980 − 0.194i)8-s + (0.231 + 0.400i)9-s + (−0.554 − 0.110i)10-s + (0.122 + 0.455i)11-s + (−0.959 + 0.735i)12-s + (−0.565 + 0.824i)13-s + (0.549 + 1.61i)14-s + (0.660 − 0.176i)15-s + (−0.706 − 0.707i)16-s + (−0.0117 + 0.00678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.873 + 0.487i$
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: \((2,\ 52,\ (\ :1/2),\ 0.873 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762252 - 0.198358i\)
\(L(\frac12)\) \(\approx\) \(0.762252 - 0.198358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.785 + 1.17i)T \)
13 \( 1 + (2.03 - 2.97i)T \)
good3 \( 1 + (-1.81 - 1.04i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.894 + 0.894i)T - 5iT^{2} \)
7 \( 1 + (4.37 + 1.17i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.404 - 1.51i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.0484 - 0.0279i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.576 + 2.15i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.528 + 0.916i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.67 + 6.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.52 - 4.52i)T + 31iT^{2} \)
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 \( 1 + (1.04 + 1.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.38 + 5.38i)T - 47iT^{2} \)
53 \( 1 + 4.40T + 53T^{2} \)
59 \( 1 + (-8.67 - 2.32i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.00 - 1.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.86 - 2.37i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.76 - 6.59i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.45 + 5.45i)T + 73iT^{2} \)
79 \( 1 - 8.87iT - 79T^{2} \)
83 \( 1 + (6.96 + 6.96i)T + 83iT^{2} \)
89 \( 1 + (15.3 - 4.10i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.440 + 0.118i)T + (84.0 + 48.5i)T^{2} \)
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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

−15.54433412417874683486912512767, −13.95573843494367560677923269885, −13.15976401624375866680724228345, −12.00339874167604896681260011201, −10.11974147526484524154679699597, −9.591144503226069274439870791140, −8.738662880075069115703941418889, −6.97443657324477593512545508829, −4.20077156306306909477527504859, −2.81817418389956831055810737892, 2.87863706854200865098661847824, 5.88702891072450094634372325147, 6.99605559038163146245281783746, 8.278414911080959161243295541406, 9.387122187686731662578187267495, 10.33464495933526671157468171685, 12.62217567409295040489947018505, 13.60420173555266287618016802527, 14.46867343584851171080534184853, 15.57090260265433162284115744069

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