Properties

Label 2-538-269.100-c1-0-1
Degree $2$
Conductor $538$
Sign $-0.934 - 0.356i$
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 + (−2.06 − 0.0483i)3-s + (−0.930 − 0.366i)4-s + (−0.765 − 2.90i)5-s + (0.431 − 2.01i)6-s + (0.890 − 1.98i)7-s + (0.533 − 0.845i)8-s + (1.24 + 0.0584i)9-s + (2.99 − 0.210i)10-s + (−0.647 + 0.727i)11-s + (1.89 + 0.799i)12-s + (0.223 + 3.17i)13-s + (1.78 + 1.24i)14-s + (1.43 + 6.01i)15-s + (0.731 + 0.681i)16-s + (−4.38 + 3.89i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (−1.18 − 0.0278i)3-s + (−0.465 − 0.183i)4-s + (−0.342 − 1.29i)5-s + (0.176 − 0.822i)6-s + (0.336 − 0.749i)7-s + (0.188 − 0.299i)8-s + (0.415 + 0.0194i)9-s + (0.947 − 0.0667i)10-s + (−0.195 + 0.219i)11-s + (0.548 + 0.230i)12-s + (0.0620 + 0.881i)13-s + (0.476 + 0.332i)14-s + (0.371 + 1.55i)15-s + (0.182 + 0.170i)16-s + (−1.06 + 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.934 - 0.356i$
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: \((2,\ 538,\ (\ :1/2),\ -0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0249287 + 0.135341i\)
\(L(\frac12)\) \(\approx\) \(0.0249287 + 0.135341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (15.4 + 5.60i)T \)
good3 \( 1 + (2.06 + 0.0483i)T + (2.99 + 0.140i)T^{2} \)
5 \( 1 + (0.765 + 2.90i)T + (-4.34 + 2.46i)T^{2} \)
7 \( 1 + (-0.890 + 1.98i)T + (-4.65 - 5.23i)T^{2} \)
11 \( 1 + (0.647 - 0.727i)T + (-1.28 - 10.9i)T^{2} \)
13 \( 1 + (-0.223 - 3.17i)T + (-12.8 + 1.82i)T^{2} \)
17 \( 1 + (4.38 - 3.89i)T + (1.98 - 16.8i)T^{2} \)
19 \( 1 + (1.29 - 0.857i)T + (7.37 - 17.5i)T^{2} \)
23 \( 1 + (-0.0751 + 3.20i)T + (-22.9 - 1.07i)T^{2} \)
29 \( 1 + (2.61 - 4.60i)T + (-14.8 - 24.8i)T^{2} \)
31 \( 1 + (2.84 - 0.335i)T + (30.1 - 7.20i)T^{2} \)
37 \( 1 + (1.28 + 2.39i)T + (-20.4 + 30.8i)T^{2} \)
41 \( 1 + (2.32 - 0.440i)T + (38.1 - 15.0i)T^{2} \)
43 \( 1 + (-4.44 - 2.52i)T + (22.0 + 36.8i)T^{2} \)
47 \( 1 + (0.272 - 0.0788i)T + (39.7 - 25.0i)T^{2} \)
53 \( 1 + (7.98 - 10.8i)T + (-15.9 - 50.5i)T^{2} \)
59 \( 1 + (1.59 - 4.05i)T + (-43.1 - 40.2i)T^{2} \)
61 \( 1 + (6.91 - 8.54i)T + (-12.7 - 59.6i)T^{2} \)
67 \( 1 + (-2.68 + 1.05i)T + (49.0 - 45.6i)T^{2} \)
71 \( 1 + (6.75 + 9.67i)T + (-24.4 + 66.6i)T^{2} \)
73 \( 1 + (-0.149 - 0.406i)T + (-55.6 + 47.2i)T^{2} \)
79 \( 1 + (1.94 - 1.50i)T + (20.1 - 76.3i)T^{2} \)
83 \( 1 + (0.804 - 8.54i)T + (-81.5 - 15.4i)T^{2} \)
89 \( 1 + (-0.744 + 4.49i)T + (-84.2 - 28.6i)T^{2} \)
97 \( 1 + (-6.95 - 8.60i)T + (-20.3 + 94.8i)T^{2} \)
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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

−11.04268486670315221085120683192, −10.58467207024498496898738264254, −9.172762296013599672881691532517, −8.581080355594606761400401526056, −7.52988963828064339581071034240, −6.58101763522647673293673202024, −5.70296782862142400267808186694, −4.63285235947857113174186276613, −4.25804613242130860304309800052, −1.39240794649207275132172270612, 0.10192917759816884697659065899, 2.34113347458488596798725849391, 3.36978791787020546121287676388, 4.84695977152715975099902385786, 5.71607233314443666787823721410, 6.67272522681332599868755867190, 7.68866392861177054013052489512, 8.780894744811539387548355578483, 9.911700947055786357085789839542, 10.79766353062664221048410368796

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