Properties

Label 2-338-13.4-c3-0-33
Degree 22
Conductor 338338
Sign 0.0841+0.996i0.0841 + 0.996i
Analytic cond. 19.942619.9426
Root an. cond. 4.465714.46571
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + i)2-s + (0.622 − 1.07i)3-s + (1.99 + 3.46i)4-s − 20.5i·5-s + (2.15 − 1.24i)6-s + (18.4 − 10.6i)7-s + 7.99i·8-s + (12.7 + 22.0i)9-s + (20.5 − 35.5i)10-s + (−45.9 − 26.5i)11-s + 4.98·12-s + 42.5·14-s + (−22.1 − 12.7i)15-s + (−8 + 13.8i)16-s + (−34.6 − 59.9i)17-s + 50.8i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.119 − 0.207i)3-s + (0.249 + 0.433i)4-s − 1.83i·5-s + (0.146 − 0.0847i)6-s + (0.993 − 0.573i)7-s + 0.353i·8-s + (0.471 + 0.816i)9-s + (0.648 − 1.12i)10-s + (−1.25 − 0.727i)11-s + 0.119·12-s + 0.811·14-s + (−0.380 − 0.219i)15-s + (−0.125 + 0.216i)16-s + (−0.493 − 0.855i)17-s + 0.666i·18-s + ⋯

Functional equation

Λ(s)=(338s/2ΓC(s)L(s)=((0.0841+0.996i)Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2ΓC(s+3/2)L(s)=((0.0841+0.996i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    21322 \cdot 13^{2}
Sign: 0.0841+0.996i0.0841 + 0.996i
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ338(147,)\chi_{338} (147, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :3/2), 0.0841+0.996i)(2,\ 338,\ (\ :3/2),\ 0.0841 + 0.996i)

Particular Values

L(2)L(2) \approx 2.6798430782.679843078
L(12)L(\frac12) \approx 2.6798430782.679843078
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.73i)T 1 + (-1.73 - i)T
13 1 1
good3 1+(0.622+1.07i)T+(13.523.3i)T2 1 + (-0.622 + 1.07i)T + (-13.5 - 23.3i)T^{2}
5 1+20.5iT125T2 1 + 20.5iT - 125T^{2}
7 1+(18.4+10.6i)T+(171.5297.i)T2 1 + (-18.4 + 10.6i)T + (171.5 - 297. i)T^{2}
11 1+(45.9+26.5i)T+(665.5+1.15e3i)T2 1 + (45.9 + 26.5i)T + (665.5 + 1.15e3i)T^{2}
17 1+(34.6+59.9i)T+(2.45e3+4.25e3i)T2 1 + (34.6 + 59.9i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(39.8+23.0i)T+(3.42e35.94e3i)T2 1 + (-39.8 + 23.0i)T + (3.42e3 - 5.94e3i)T^{2}
23 1+(43.675.5i)T+(6.08e31.05e4i)T2 1 + (43.6 - 75.5i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(81.0+140.i)T+(1.21e42.11e4i)T2 1 + (-81.0 + 140. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+28.3iT2.97e4T2 1 + 28.3iT - 2.97e4T^{2}
37 1+(96.9+55.9i)T+(2.53e4+4.38e4i)T2 1 + (96.9 + 55.9i)T + (2.53e4 + 4.38e4i)T^{2}
41 1+(72.942.1i)T+(3.44e4+5.96e4i)T2 1 + (-72.9 - 42.1i)T + (3.44e4 + 5.96e4i)T^{2}
43 1+(164.+284.i)T+(3.97e4+6.88e4i)T2 1 + (164. + 284. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 163.2iT1.03e5T2 1 - 63.2iT - 1.03e5T^{2}
53 1721.T+1.48e5T2 1 - 721.T + 1.48e5T^{2}
59 1+(709.+409.i)T+(1.02e51.77e5i)T2 1 + (-709. + 409. i)T + (1.02e5 - 1.77e5i)T^{2}
61 1+(198.344.i)T+(1.13e5+1.96e5i)T2 1 + (-198. - 344. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(67.4+38.9i)T+(1.50e5+2.60e5i)T2 1 + (67.4 + 38.9i)T + (1.50e5 + 2.60e5i)T^{2}
71 1+(624.+360.i)T+(1.78e53.09e5i)T2 1 + (-624. + 360. i)T + (1.78e5 - 3.09e5i)T^{2}
73 157.1iT3.89e5T2 1 - 57.1iT - 3.89e5T^{2}
79 1+419.T+4.93e5T2 1 + 419.T + 4.93e5T^{2}
83 1917.iT5.71e5T2 1 - 917. iT - 5.71e5T^{2}
89 1+(328.+189.i)T+(3.52e5+6.10e5i)T2 1 + (328. + 189. i)T + (3.52e5 + 6.10e5i)T^{2}
97 1+(300.173.i)T+(4.56e57.90e5i)T2 1 + (300. - 173. i)T + (4.56e5 - 7.90e5i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.15386940930845902546945051422, −9.958530472859566779553353352669, −8.612893949473297202781168138292, −8.036190373266643370288401685550, −7.31221906248212515468418099179, −5.44662039614328278535923451922, −5.03807172710048136000385529281, −4.13657438736487968517915652246, −2.18474614709589689505723538942, −0.75850515719202127173885847193, 1.99661022148926967086311631639, 2.93204603150945446695540407121, 4.09101468856274269720556848912, 5.33993645813843285430740969798, 6.49614208368045295120230608296, 7.30138394418491732743404718010, 8.445953302868345638747281993313, 10.03873149985349852398555174158, 10.40109520433923868808933617512, 11.30754623132320200797405910138

Graph of the ZZ-function along the critical line