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Properties

Label 2-338-13.10-c3-0-4
Degree 22
Conductor 338338
Sign −0.711−0.702i-0.711 - 0.702i
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 + (−2.93 − 5.07i)3-s + (1.99 − 3.46i)4-s + 10.8i·5-s + (10.1 + 5.86i)6-s + (25.8 + 14.9i)7-s + 7.99i·8-s + (−3.70 + 6.41i)9-s + (−10.8 − 18.8i)10-s + (−42.4 + 24.5i)11-s − 23.4·12-s − 59.7·14-s + (55.1 − 31.8i)15-s + (−8 − 13.8i)16-s + (−3.03 + 5.26i)17-s − 14.8i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.564 − 0.977i)3-s + (0.249 − 0.433i)4-s + 0.971i·5-s + (0.691 + 0.399i)6-s + (1.39 + 0.806i)7-s + 0.353i·8-s + (−0.137 + 0.237i)9-s + (−0.343 − 0.595i)10-s + (−1.16 + 0.672i)11-s − 0.564·12-s − 1.14·14-s + (0.950 − 0.548i)15-s + (−0.125 − 0.216i)16-s + (−0.0433 + 0.0750i)17-s − 0.193i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((−0.711−0.702i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=((−0.711−0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: −0.711−0.702i-0.711 - 0.702i
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ338(23,⋅)\chi_{338} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :3/2), −0.711−0.702i)(2,\ 338,\ (\ :3/2),\ -0.711 - 0.702i)

Particular Values

L(2)L(2) ≈\approx 0.60429724600.6042972460
L(12)L(\frac12) ≈\approx 0.60429724600.6042972460
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+(1.73−i)T 1 + (1.73 - i)T
13 1 1
good3 1+(2.93+5.07i)T+(−13.5+23.3i)T2 1 + (2.93 + 5.07i)T + (-13.5 + 23.3i)T^{2}
5 1−10.8iT−125T2 1 - 10.8iT - 125T^{2}
7 1+(−25.8−14.9i)T+(171.5+297.i)T2 1 + (-25.8 - 14.9i)T + (171.5 + 297. i)T^{2}
11 1+(42.4−24.5i)T+(665.5−1.15e3i)T2 1 + (42.4 - 24.5i)T + (665.5 - 1.15e3i)T^{2}
17 1+(3.03−5.26i)T+(−2.45e3−4.25e3i)T2 1 + (3.03 - 5.26i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(5.07+2.93i)T+(3.42e3+5.94e3i)T2 1 + (5.07 + 2.93i)T + (3.42e3 + 5.94e3i)T^{2}
23 1+(2.79+4.84i)T+(−6.08e3+1.05e4i)T2 1 + (2.79 + 4.84i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−5.30−9.19i)T+(−1.21e4+2.11e4i)T2 1 + (-5.30 - 9.19i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+316.iT−2.97e4T2 1 + 316. iT - 2.97e4T^{2}
37 1+(329.−190.i)T+(2.53e4−4.38e4i)T2 1 + (329. - 190. i)T + (2.53e4 - 4.38e4i)T^{2}
41 1+(232.−134.i)T+(3.44e4−5.96e4i)T2 1 + (232. - 134. i)T + (3.44e4 - 5.96e4i)T^{2}
43 1+(115.−199.i)T+(−3.97e4−6.88e4i)T2 1 + (115. - 199. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1−524.iT−1.03e5T2 1 - 524. iT - 1.03e5T^{2}
53 1+274.T+1.48e5T2 1 + 274.T + 1.48e5T^{2}
59 1+(−123.−71.4i)T+(1.02e5+1.77e5i)T2 1 + (-123. - 71.4i)T + (1.02e5 + 1.77e5i)T^{2}
61 1+(281.−487.i)T+(−1.13e5−1.96e5i)T2 1 + (281. - 487. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−448.+258.i)T+(1.50e5−2.60e5i)T2 1 + (-448. + 258. i)T + (1.50e5 - 2.60e5i)T^{2}
71 1+(125.+72.2i)T+(1.78e5+3.09e5i)T2 1 + (125. + 72.2i)T + (1.78e5 + 3.09e5i)T^{2}
73 1+201.iT−3.89e5T2 1 + 201. iT - 3.89e5T^{2}
79 1−26.4T+4.93e5T2 1 - 26.4T + 4.93e5T^{2}
83 1−1.14e3iT−5.71e5T2 1 - 1.14e3iT - 5.71e5T^{2}
89 1+(574.−331.i)T+(3.52e5−6.10e5i)T2 1 + (574. - 331. i)T + (3.52e5 - 6.10e5i)T^{2}
97 1+(119.+68.7i)T+(4.56e5+7.90e5i)T2 1 + (119. + 68.7i)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

−11.37690849673476749156409897922, −10.71223312194888490670458260373, −9.642771519617851776243009953700, −8.249900850935379421073010405436, −7.67489688508020098937801761552, −6.80798478120333860250745644543, −5.85912776532393590769076584373, −4.85933886771107384621629355604, −2.56869791771865697359717370465, −1.56438903509914293313333928009, 0.28402074319882111555355107042, 1.68513244868644053956339535474, 3.67154642900045509086949414952, 4.94161860972544974853277685720, 5.23961519272080087661580952661, 7.20440514524411438138712463491, 8.266993607448287654165908902305, 8.804599725498628315869561067805, 10.23017471616967562864269975196, 10.60294341017151400034548232749

Graph of the ZZ-function along the critical line