Properties

Label 2-538-269.100-c1-0-17
Degree $2$
Conductor $538$
Sign $0.973 + 0.229i$
Analytic cond. $4.29595$
Root an. cond. $2.07266$
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  + (−0.186 + 0.982i)2-s + (0.932 + 0.0218i)3-s + (−0.930 − 0.366i)4-s + (0.272 + 1.03i)5-s + (−0.195 + 0.912i)6-s + (1.94 − 4.33i)7-s + (0.533 − 0.845i)8-s + (−2.12 − 0.0998i)9-s + (−1.06 + 0.0752i)10-s + (3.06 − 3.44i)11-s + (−0.860 − 0.362i)12-s + (−0.431 − 6.12i)13-s + (3.89 + 2.72i)14-s + (0.232 + 0.971i)15-s + (0.731 + 0.681i)16-s + (−3.31 + 2.94i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (0.538 + 0.0126i)3-s + (−0.465 − 0.183i)4-s + (0.122 + 0.462i)5-s + (−0.0797 + 0.372i)6-s + (0.735 − 1.63i)7-s + (0.188 − 0.299i)8-s + (−0.708 − 0.0332i)9-s + (−0.337 + 0.0237i)10-s + (0.922 − 1.03i)11-s + (−0.248 − 0.104i)12-s + (−0.119 − 1.69i)13-s + (1.04 + 0.727i)14-s + (0.0599 + 0.250i)15-s + (0.182 + 0.170i)16-s + (−0.802 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(4.29595\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55797 - 0.181133i\)
\(L(\frac12)\) \(\approx\) \(1.55797 - 0.181133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.186 - 0.982i)T \)
269 \( 1 + (-16.0 - 3.51i)T \)
good3 \( 1 + (-0.932 - 0.0218i)T + (2.99 + 0.140i)T^{2} \)
5 \( 1 + (-0.272 - 1.03i)T + (-4.34 + 2.46i)T^{2} \)
7 \( 1 + (-1.94 + 4.33i)T + (-4.65 - 5.23i)T^{2} \)
11 \( 1 + (-3.06 + 3.44i)T + (-1.28 - 10.9i)T^{2} \)
13 \( 1 + (0.431 + 6.12i)T + (-12.8 + 1.82i)T^{2} \)
17 \( 1 + (3.31 - 2.94i)T + (1.98 - 16.8i)T^{2} \)
19 \( 1 + (-1.25 + 0.831i)T + (7.37 - 17.5i)T^{2} \)
23 \( 1 + (0.0343 - 1.46i)T + (-22.9 - 1.07i)T^{2} \)
29 \( 1 + (4.24 - 7.48i)T + (-14.8 - 24.8i)T^{2} \)
31 \( 1 + (-3.16 + 0.372i)T + (30.1 - 7.20i)T^{2} \)
37 \( 1 + (-3.72 - 6.95i)T + (-20.4 + 30.8i)T^{2} \)
41 \( 1 + (1.94 - 0.368i)T + (38.1 - 15.0i)T^{2} \)
43 \( 1 + (-2.13 - 1.21i)T + (22.0 + 36.8i)T^{2} \)
47 \( 1 + (-9.96 + 2.87i)T + (39.7 - 25.0i)T^{2} \)
53 \( 1 + (5.13 - 7.00i)T + (-15.9 - 50.5i)T^{2} \)
59 \( 1 + (-1.32 + 3.36i)T + (-43.1 - 40.2i)T^{2} \)
61 \( 1 + (-5.85 + 7.23i)T + (-12.7 - 59.6i)T^{2} \)
67 \( 1 + (2.45 - 0.966i)T + (49.0 - 45.6i)T^{2} \)
71 \( 1 + (-3.25 - 4.66i)T + (-24.4 + 66.6i)T^{2} \)
73 \( 1 + (-3.48 - 9.49i)T + (-55.6 + 47.2i)T^{2} \)
79 \( 1 + (2.89 - 2.22i)T + (20.1 - 76.3i)T^{2} \)
83 \( 1 + (1.59 - 16.9i)T + (-81.5 - 15.4i)T^{2} \)
89 \( 1 + (-1.09 + 6.60i)T + (-84.2 - 28.6i)T^{2} \)
97 \( 1 + (-0.866 - 1.07i)T + (-20.3 + 94.8i)T^{2} \)
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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

−10.84885147835766412937316808385, −9.896014843895020532362659384752, −8.662602886100140163300788377982, −8.154696757542749368890199779753, −7.27529428921014756817024427639, −6.36668631939220063324033012079, −5.30207440443958534585414153450, −3.99842176828936225943741763163, −3.09438439911888905429143912854, −0.969064439609761032719147990020, 1.91111587342730472084300820732, 2.45434203971192553439981055752, 4.15742498510686828762704133521, 4.98951943434619601423406850163, 6.15547444704976669579265884475, 7.49130305566079557105451755453, 8.702844788240429849466397082896, 9.116663730968845359093888466719, 9.513581416904266318058227832198, 11.20115283761584145284936982606

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