↑

Properties

Label 2-338-13.4-c3-0-19
Degree 22
Conductor 338338
Sign 0.327−0.944i0.327 - 0.944i
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  + (1.73 + i)2-s + (−1.67 + 2.90i)3-s + (1.99 + 3.46i)4-s − 6.80i·5-s + (−5.81 + 3.35i)6-s + (13.4 − 7.76i)7-s + 7.99i·8-s + (7.86 + 13.6i)9-s + (6.80 − 11.7i)10-s + (−16.9 − 9.80i)11-s − 13.4·12-s + 31.0·14-s + (19.7 + 11.4i)15-s + (−8 + 13.8i)16-s + (62.9 + 109. i)17-s + 31.4i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.323 + 0.559i)3-s + (0.249 + 0.433i)4-s − 0.608i·5-s + (−0.395 + 0.228i)6-s + (0.726 − 0.419i)7-s + 0.353i·8-s + (0.291 + 0.504i)9-s + (0.215 − 0.372i)10-s + (−0.465 − 0.268i)11-s − 0.323·12-s + 0.593·14-s + (0.340 + 0.196i)15-s + (−0.125 + 0.216i)16-s + (0.898 + 1.55i)17-s + 0.411i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.327−0.944i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=((0.327−0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.327−0.944i0.327 - 0.944i
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), 0.327−0.944i)(2,\ 338,\ (\ :3/2),\ 0.327 - 0.944i)

Particular Values

L(2)L(2) ≈\approx 2.6818741002.681874100
L(12)L(\frac12) ≈\approx 2.6818741002.681874100
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+(1.67−2.90i)T+(−13.5−23.3i)T2 1 + (1.67 - 2.90i)T + (-13.5 - 23.3i)T^{2}
5 1+6.80iT−125T2 1 + 6.80iT - 125T^{2}
7 1+(−13.4+7.76i)T+(171.5−297.i)T2 1 + (-13.4 + 7.76i)T + (171.5 - 297. i)T^{2}
11 1+(16.9+9.80i)T+(665.5+1.15e3i)T2 1 + (16.9 + 9.80i)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+(−84.1+48.6i)T+(3.42e3−5.94e3i)T2 1 + (-84.1 + 48.6i)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.iT−2.97e4T2 1 - 172. iT - 2.97e4T^{2}
37 1+(−191.−110.i)T+(2.53e4+4.38e4i)T2 1 + (-191. - 110. i)T + (2.53e4 + 4.38e4i)T^{2}
41 1+(24.3+14.0i)T+(3.44e4+5.96e4i)T2 1 + (24.3 + 14.0i)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.iT−1.03e5T2 1 - 456. iT - 1.03e5T^{2}
53 1+643.T+1.48e5T2 1 + 643.T + 1.48e5T^{2}
59 1+(−269.+155.i)T+(1.02e5−1.77e5i)T2 1 + (-269. + 155. 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+(−6.06−3.50i)T+(1.50e5+2.60e5i)T2 1 + (-6.06 - 3.50i)T + (1.50e5 + 2.60e5i)T^{2}
71 1+(−736.+425.i)T+(1.78e5−3.09e5i)T2 1 + (-736. + 425. i)T + (1.78e5 - 3.09e5i)T^{2}
73 1+1.12e3iT−3.89e5T2 1 + 1.12e3iT - 3.89e5T^{2}
79 1−278.T+4.93e5T2 1 - 278.T + 4.93e5T^{2}
83 1−417.iT−5.71e5T2 1 - 417. iT - 5.71e5T^{2}
89 1+(−280.−162.i)T+(3.52e5+6.10e5i)T2 1 + (-280. - 162. i)T + (3.52e5 + 6.10e5i)T^{2}
97 1+(−71.0+41.0i)T+(4.56e5−7.90e5i)T2 1 + (-71.0 + 41.0i)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

−11.01566746393021574944688777372, −10.76969387573599107564824999409, −9.461127340459413334730313173173, −8.225064051695225223232055007670, −7.60768604941582533189743618566, −6.21020874832795463732895511413, −4.98859613041482958180860376489, −4.68492105100075040462294568303, −3.25629251016813230089408409566, −1.35229409589158269901490038006, 0.962249725818217175479211349815, 2.38676607133104433079549870171, 3.59428450099678319233873907574, 5.10345411777553571916590782244, 5.83059124333673250204245430200, 7.15720978689615040636471966046, 7.65737349148508557176558189504, 9.340543176082677203312956612608, 10.09655106469691579910482462853, 11.49836030831675194919277766192

Graph of the ZZ-function along the critical line