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Properties

Label 2-538-269.191-c1-0-4
Degree 22
Conductor 538538
Sign 0.223−0.974i0.223 - 0.974i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.186 + 0.982i)2-s + (−2.89 + 0.0679i)3-s + (−0.930 + 0.366i)4-s + (0.352 − 1.33i)5-s + (−0.607 − 2.83i)6-s + (1.01 + 2.26i)7-s + (−0.533 − 0.845i)8-s + (5.39 − 0.253i)9-s + (1.37 + 0.0971i)10-s + (−2.76 − 3.10i)11-s + (2.67 − 1.12i)12-s + (0.287 − 4.07i)13-s + (−2.03 + 1.42i)14-s + (−0.930 + 3.89i)15-s + (0.731 − 0.681i)16-s + (4.51 + 4.01i)17-s + ⋯
L(s)  = 1  + (0.131 + 0.694i)2-s + (−1.67 + 0.0392i)3-s + (−0.465 + 0.183i)4-s + (0.157 − 0.597i)5-s + (−0.247 − 1.15i)6-s + (0.384 + 0.856i)7-s + (−0.188 − 0.299i)8-s + (1.79 − 0.0843i)9-s + (0.436 + 0.0307i)10-s + (−0.832 − 0.935i)11-s + (0.771 − 0.324i)12-s + (0.0796 − 1.13i)13-s + (−0.544 + 0.379i)14-s + (−0.240 + 1.00i)15-s + (0.182 − 0.170i)16-s + (1.09 + 0.973i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.223−0.974i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((0.223−0.974i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.223−0.974i0.223 - 0.974i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(191,⋅)\chi_{538} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 0.223−0.974i)(2,\ 538,\ (\ :1/2),\ 0.223 - 0.974i)

Particular Values

L(1)L(1) ≈\approx 0.643472+0.512662i0.643472 + 0.512662i
L(12)L(\frac12) ≈\approx 0.643472+0.512662i0.643472 + 0.512662i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.186−0.982i)T 1 + (-0.186 - 0.982i)T
269 1+(3.91−15.9i)T 1 + (3.91 - 15.9i)T
good3 1+(2.89−0.0679i)T+(2.99−0.140i)T2 1 + (2.89 - 0.0679i)T + (2.99 - 0.140i)T^{2}
5 1+(−0.352+1.33i)T+(−4.34−2.46i)T2 1 + (-0.352 + 1.33i)T + (-4.34 - 2.46i)T^{2}
7 1+(−1.01−2.26i)T+(−4.65+5.23i)T2 1 + (-1.01 - 2.26i)T + (-4.65 + 5.23i)T^{2}
11 1+(2.76+3.10i)T+(−1.28+10.9i)T2 1 + (2.76 + 3.10i)T + (-1.28 + 10.9i)T^{2}
13 1+(−0.287+4.07i)T+(−12.8−1.82i)T2 1 + (-0.287 + 4.07i)T + (-12.8 - 1.82i)T^{2}
17 1+(−4.51−4.01i)T+(1.98+16.8i)T2 1 + (-4.51 - 4.01i)T + (1.98 + 16.8i)T^{2}
19 1+(−1.36−0.905i)T+(7.37+17.5i)T2 1 + (-1.36 - 0.905i)T + (7.37 + 17.5i)T^{2}
23 1+(−0.169−7.21i)T+(−22.9+1.07i)T2 1 + (-0.169 - 7.21i)T + (-22.9 + 1.07i)T^{2}
29 1+(−4.50−7.94i)T+(−14.8+24.8i)T2 1 + (-4.50 - 7.94i)T + (-14.8 + 24.8i)T^{2}
31 1+(9.49+1.11i)T+(30.1+7.20i)T2 1 + (9.49 + 1.11i)T + (30.1 + 7.20i)T^{2}
37 1+(3.22−6.00i)T+(−20.4−30.8i)T2 1 + (3.22 - 6.00i)T + (-20.4 - 30.8i)T^{2}
41 1+(−2.25−0.428i)T+(38.1+15.0i)T2 1 + (-2.25 - 0.428i)T + (38.1 + 15.0i)T^{2}
43 1+(−9.34+5.29i)T+(22.0−36.8i)T2 1 + (-9.34 + 5.29i)T + (22.0 - 36.8i)T^{2}
47 1+(−12.5−3.62i)T+(39.7+25.0i)T2 1 + (-12.5 - 3.62i)T + (39.7 + 25.0i)T^{2}
53 1+(6.57+8.96i)T+(−15.9+50.5i)T2 1 + (6.57 + 8.96i)T + (-15.9 + 50.5i)T^{2}
59 1+(−1.00−2.56i)T+(−43.1+40.2i)T2 1 + (-1.00 - 2.56i)T + (-43.1 + 40.2i)T^{2}
61 1+(−4.14−5.12i)T+(−12.7+59.6i)T2 1 + (-4.14 - 5.12i)T + (-12.7 + 59.6i)T^{2}
67 1+(−1.35−0.533i)T+(49.0+45.6i)T2 1 + (-1.35 - 0.533i)T + (49.0 + 45.6i)T^{2}
71 1+(2.03−2.91i)T+(−24.4−66.6i)T2 1 + (2.03 - 2.91i)T + (-24.4 - 66.6i)T^{2}
73 1+(−5.00+13.6i)T+(−55.6−47.2i)T2 1 + (-5.00 + 13.6i)T + (-55.6 - 47.2i)T^{2}
79 1+(−4.33−3.33i)T+(20.1+76.3i)T2 1 + (-4.33 - 3.33i)T + (20.1 + 76.3i)T^{2}
83 1+(−0.720−7.65i)T+(−81.5+15.4i)T2 1 + (-0.720 - 7.65i)T + (-81.5 + 15.4i)T^{2}
89 1+(−0.527−3.18i)T+(−84.2+28.6i)T2 1 + (-0.527 - 3.18i)T + (-84.2 + 28.6i)T^{2}
97 1+(−4.61+5.71i)T+(−20.3−94.8i)T2 1 + (-4.61 + 5.71i)T + (-20.3 - 94.8i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.92626741333792661963587262641, −10.41067497707184099797885914283, −9.147749261117213314901354018199, −8.202007268974027237538878902686, −7.32681399955157112602132420158, −5.93077585816966600062422688497, −5.40920902521538221353241333663, −5.22564642929301767950024890015, −3.47603948128307781128832311765, −1.08139959235657882790868148349, 0.74393771154463259705902975382, 2.38342015592441755923376098216, 4.21370202929193803151271595231, 4.85222356176392932884965040297, 5.87593438799176735092848974465, 6.96786118343972421841294489646, 7.54980812644120932333891125549, 9.326922402531362678439619267234, 10.26983168894997568566815956092, 10.70970275598349266916517189725

Graph of the ZZ-function along the critical line