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Properties

Label 2-338-13.9-c3-0-12
Degree 22
Conductor 338338
Sign 0.611−0.791i0.611 - 0.791i
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 + 1.73i)2-s + (−5.00 + 8.66i)3-s + (−1.99 − 3.46i)4-s − 13.8·5-s + (−10.0 − 17.3i)6-s + (0.131 + 0.227i)7-s + 7.99·8-s + (−36.5 − 63.3i)9-s + (13.8 − 23.9i)10-s + (−20.0 + 34.6i)11-s + 40.0·12-s − 0.524·14-s + (69.1 − 119. i)15-s + (−8 + 13.8i)16-s + (−39.8 − 69.0i)17-s + 146.·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.963 + 1.66i)3-s + (−0.249 − 0.433i)4-s − 1.23·5-s + (−0.680 − 1.17i)6-s + (0.00708 + 0.0122i)7-s + 0.353·8-s + (−1.35 − 2.34i)9-s + (0.436 − 0.756i)10-s + (−0.548 + 0.949i)11-s + 0.963·12-s − 0.0100·14-s + (1.18 − 2.06i)15-s + (−0.125 + 0.216i)16-s + (−0.568 − 0.985i)17-s + 1.91·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.611−0.791i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=((0.611−0.791i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.611−0.791i0.611 - 0.791i
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ338(191,⋅)\chi_{338} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :3/2), 0.611−0.791i)(2,\ 338,\ (\ :3/2),\ 0.611 - 0.791i)

Particular Values

L(2)L(2) ≈\approx 0.27273911370.2727391137
L(12)L(\frac12) ≈\approx 0.27273911370.2727391137
L(52)L(\frac{5}{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+(1−1.73i)T 1 + (1 - 1.73i)T
13 1 1
good3 1+(5.00−8.66i)T+(−13.5−23.3i)T2 1 + (5.00 - 8.66i)T + (-13.5 - 23.3i)T^{2}
5 1+13.8T+125T2 1 + 13.8T + 125T^{2}
7 1+(−0.131−0.227i)T+(−171.5+297.i)T2 1 + (-0.131 - 0.227i)T + (-171.5 + 297. i)T^{2}
11 1+(20.0−34.6i)T+(−665.5−1.15e3i)T2 1 + (20.0 - 34.6i)T + (-665.5 - 1.15e3i)T^{2}
17 1+(39.8+69.0i)T+(−2.45e3+4.25e3i)T2 1 + (39.8 + 69.0i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(−11.2−19.5i)T+(−3.42e3+5.94e3i)T2 1 + (-11.2 - 19.5i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−32.7+56.7i)T+(−6.08e3−1.05e4i)T2 1 + (-32.7 + 56.7i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(20.0−34.7i)T+(−1.21e4−2.11e4i)T2 1 + (20.0 - 34.7i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+113.T+2.97e4T2 1 + 113.T + 2.97e4T^{2}
37 1+(60.5−104.i)T+(−2.53e4−4.38e4i)T2 1 + (60.5 - 104. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+(198.−343.i)T+(−3.44e4−5.96e4i)T2 1 + (198. - 343. i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(−137.−238.i)T+(−3.97e4+6.88e4i)T2 1 + (-137. - 238. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+440.T+1.03e5T2 1 + 440.T + 1.03e5T^{2}
53 1+615.T+1.48e5T2 1 + 615.T + 1.48e5T^{2}
59 1+(−115.−199.i)T+(−1.02e5+1.77e5i)T2 1 + (-115. - 199. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−54.7−94.8i)T+(−1.13e5+1.96e5i)T2 1 + (-54.7 - 94.8i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−110.+191.i)T+(−1.50e5−2.60e5i)T2 1 + (-110. + 191. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+(−161.−279.i)T+(−1.78e5+3.09e5i)T2 1 + (-161. - 279. i)T + (-1.78e5 + 3.09e5i)T^{2}
73 1+323.T+3.89e5T2 1 + 323.T + 3.89e5T^{2}
79 1−743.T+4.93e5T2 1 - 743.T + 4.93e5T^{2}
83 1−539.T+5.71e5T2 1 - 539.T + 5.71e5T^{2}
89 1+(−632.+1.09e3i)T+(−3.52e5−6.10e5i)T2 1 + (-632. + 1.09e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+(163.+282.i)T+(−4.56e5+7.90e5i)T2 1 + (163. + 282. i)T + (-4.56e5 + 7.90e5i)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

−11.20558828179706041019016456171, −10.22359272260450134354508409751, −9.546018803563917546602329984076, −8.564721314692087588023115697690, −7.41440033446755901545908698119, −6.34309796833316313935361459711, −4.95684878472353706415164311882, −4.61354576574586526874163579798, −3.37202537418776790414338727218, −0.23398107623857426181468877493, 0.61157026520456943401805004059, 2.00578520986497661156765611700, 3.51942958877205766270335195711, 5.13573600330330407670012301788, 6.28112027910182137160925939962, 7.37491140334800760700501252003, 7.965683845798451598853938680737, 8.770802337384516801531145246427, 10.70618873762043459410769710204, 11.12243141640079619532216087920

Graph of the ZZ-function along the critical line