↑

Properties

Label 2-338-13.3-c3-0-35
Degree 22
Conductor 338338
Sign −0.562−0.826i-0.562 - 0.826i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.67 − 2.90i)3-s + (−1.99 + 3.46i)4-s + 6.80·5-s + (−3.35 + 5.81i)6-s + (−7.76 + 13.4i)7-s + 7.99·8-s + (7.86 − 13.6i)9-s + (−6.80 − 11.7i)10-s + (−9.80 − 16.9i)11-s + 13.4·12-s + 31.0·14-s + (−11.4 − 19.7i)15-s + (−8 − 13.8i)16-s + (−62.9 + 109. i)17-s − 31.4·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.323 − 0.559i)3-s + (−0.249 + 0.433i)4-s + 0.608·5-s + (−0.228 + 0.395i)6-s + (−0.419 + 0.726i)7-s + 0.353·8-s + (0.291 − 0.504i)9-s + (−0.215 − 0.372i)10-s + (−0.268 − 0.465i)11-s + 0.323·12-s + 0.593·14-s + (−0.196 − 0.340i)15-s + (−0.125 − 0.216i)16-s + (−0.898 + 1.55i)17-s − 0.411·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((−0.562−0.826i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=((−0.562−0.826i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: −0.562−0.826i-0.562 - 0.826i
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ338(315,⋅)\chi_{338} (315, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :3/2), −0.562−0.826i)(2,\ 338,\ (\ :3/2),\ -0.562 - 0.826i)

Particular Values

L(2)L(2) ≈\approx 0.11015022780.1101502278
L(12)L(\frac12) ≈\approx 0.11015022780.1101502278
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+(1.67+2.90i)T+(−13.5+23.3i)T2 1 + (1.67 + 2.90i)T + (-13.5 + 23.3i)T^{2}
5 1−6.80T+125T2 1 - 6.80T + 125T^{2}
7 1+(7.76−13.4i)T+(−171.5−297.i)T2 1 + (7.76 - 13.4i)T + (-171.5 - 297. i)T^{2}
11 1+(9.80+16.9i)T+(−665.5+1.15e3i)T2 1 + (9.80 + 16.9i)T + (-665.5 + 1.15e3i)T^{2}
17 1+(62.9−109.i)T+(−2.45e3−4.25e3i)T2 1 + (62.9 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(−48.6+84.1i)T+(−3.42e3−5.94e3i)T2 1 + (-48.6 + 84.1i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(64.9+112.i)T+(−6.08e3+1.05e4i)T2 1 + (64.9 + 112. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(73.9+128.i)T+(−1.21e4+2.11e4i)T2 1 + (73.9 + 128. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+172.T+2.97e4T2 1 + 172.T + 2.97e4T^{2}
37 1+(−110.−191.i)T+(−2.53e4+4.38e4i)T2 1 + (-110. - 191. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+(−14.0−24.3i)T+(−3.44e4+5.96e4i)T2 1 + (-14.0 - 24.3i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(180.−313.i)T+(−3.97e4−6.88e4i)T2 1 + (180. - 313. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1−456.T+1.03e5T2 1 - 456.T + 1.03e5T^{2}
53 1+643.T+1.48e5T2 1 + 643.T + 1.48e5T^{2}
59 1+(155.−269.i)T+(−1.02e5−1.77e5i)T2 1 + (155. - 269. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(405.−702.i)T+(−1.13e5−1.96e5i)T2 1 + (405. - 702. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(3.50+6.06i)T+(−1.50e5+2.60e5i)T2 1 + (3.50 + 6.06i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+(−425.+736.i)T+(−1.78e5−3.09e5i)T2 1 + (-425. + 736. i)T + (-1.78e5 - 3.09e5i)T^{2}
73 1+1.12e3T+3.89e5T2 1 + 1.12e3T + 3.89e5T^{2}
79 1−278.T+4.93e5T2 1 - 278.T + 4.93e5T^{2}
83 1+417.T+5.71e5T2 1 + 417.T + 5.71e5T^{2}
89 1+(−162.−280.i)T+(−3.52e5+6.10e5i)T2 1 + (-162. - 280. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+(−41.0+71.0i)T+(−4.56e5−7.90e5i)T2 1 + (-41.0 + 71.0i)T + (-4.56e5 - 7.90e5i)T^{2}
show more
show less
   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

−10.55570420037882767777513225729, −9.542516349291478801454044487569, −8.874584864688726719661325050938, −7.75934521789105466092377294897, −6.41727060226573768031153113172, −5.85988459107133102787513502323, −4.21885308652593982300053284370, −2.75819790416624013943584401189, −1.61285183821193435692557647651, −0.04414515973598885445615711948, 1.87286468899955480061164888706, 3.80626594586676899720621004984, 5.00987581563364782378873945088, 5.78408717750162106831263671940, 7.13292105080760607778089471467, 7.65343855409383640895052670045, 9.260422008775249771007609980278, 9.763633652341901251984564124986, 10.52516872485987633301080756559, 11.42055664625180150492615943610

Graph of the ZZ-function along the critical line