↑

Properties

Label 2-197-197.2-c2-0-20
Degree 22
Conductor 197197
Sign −0.911+0.410i-0.911 + 0.410i
Analytic cond. 5.367865.36786
Root an. cond. 2.316862.31686
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.90 − 0.0465i)2-s + (−0.438 + 0.107i)3-s + (4.41 + 0.141i)4-s + (−1.33 − 9.17i)5-s + (1.27 − 0.291i)6-s + (6.15 − 6.35i)7-s + (−1.21 − 0.0582i)8-s + (−7.79 + 4.06i)9-s + (3.44 + 26.6i)10-s + (15.1 + 7.61i)11-s + (−1.95 + 0.412i)12-s + (−17.2 − 7.29i)13-s + (−18.1 + 18.1i)14-s + (1.57 + 3.88i)15-s + (−14.1 − 0.906i)16-s + (12.9 − 8.73i)17-s + ⋯
L(s)  = 1  + (−1.45 − 0.0232i)2-s + (−0.146 + 0.0358i)3-s + (1.10 + 0.0354i)4-s + (−0.266 − 1.83i)5-s + (0.213 − 0.0486i)6-s + (0.878 − 0.907i)7-s + (−0.151 − 0.00728i)8-s + (−0.866 + 0.452i)9-s + (0.344 + 2.66i)10-s + (1.38 + 0.692i)11-s + (−0.162 + 0.0344i)12-s + (−1.32 − 0.561i)13-s + (−1.29 + 1.29i)14-s + (0.104 + 0.258i)15-s + (−0.882 − 0.0566i)16-s + (0.762 − 0.513i)17-s + ⋯

Functional equation

Λ(s)=(197s/2 ΓC(s) L(s)=((−0.911+0.410i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(197s/2 ΓC(s+1) L(s)=((−0.911+0.410i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 197197
Sign: −0.911+0.410i-0.911 + 0.410i
Analytic conductor: 5.367865.36786
Root analytic conductor: 2.316862.31686
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ197(2,⋅)\chi_{197} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 197, ( :1), −0.911+0.410i)(2,\ 197,\ (\ :1),\ -0.911 + 0.410i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.100545−0.467656i0.100545 - 0.467656i
L(12)L(\frac12) ≈\approx 0.100545−0.467656i0.100545 - 0.467656i
L(2)L(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)
bad197 1+(−77.8+180.i)T 1 + (-77.8 + 180. i)T
good2 1+(2.90+0.0465i)T+(3.99+0.128i)T2 1 + (2.90 + 0.0465i)T + (3.99 + 0.128i)T^{2}
3 1+(0.438−0.107i)T+(7.97−4.16i)T2 1 + (0.438 - 0.107i)T + (7.97 - 4.16i)T^{2}
5 1+(1.33+9.17i)T+(−23.9+7.11i)T2 1 + (1.33 + 9.17i)T + (-23.9 + 7.11i)T^{2}
7 1+(−6.15+6.35i)T+(−1.57−48.9i)T2 1 + (-6.15 + 6.35i)T + (-1.57 - 48.9i)T^{2}
11 1+(−15.1−7.61i)T+(72.3+96.9i)T2 1 + (-15.1 - 7.61i)T + (72.3 + 96.9i)T^{2}
13 1+(17.2+7.29i)T+(117.+121.i)T2 1 + (17.2 + 7.29i)T + (117. + 121. i)T^{2}
17 1+(−12.9+8.73i)T+(108.−267.i)T2 1 + (-12.9 + 8.73i)T + (108. - 267. i)T^{2}
19 1+(−4.74+3.78i)T+(80.3−351.i)T2 1 + (-4.74 + 3.78i)T + (80.3 - 351. i)T^{2}
23 1+(−6.08+16.5i)T+(−402.−342.i)T2 1 + (-6.08 + 16.5i)T + (-402. - 342. i)T^{2}
29 1+(33.2+21.6i)T+(340.+769.i)T2 1 + (33.2 + 21.6i)T + (340. + 769. i)T^{2}
31 1+(19.0−23.1i)T+(−183.−943.i)T2 1 + (19.0 - 23.1i)T + (-183. - 943. i)T^{2}
37 1+(50.1−3.21i)T+(1.35e3−175.i)T2 1 + (50.1 - 3.21i)T + (1.35e3 - 175. i)T^{2}
41 1+(−8.45+43.4i)T+(−1.55e3−630.i)T2 1 + (-8.45 + 43.4i)T + (-1.55e3 - 630. i)T^{2}
43 1+(−12.7−38.4i)T+(−1.48e3+1.10e3i)T2 1 + (-12.7 - 38.4i)T + (-1.48e3 + 1.10e3i)T^{2}
47 1+(−12.8−22.8i)T+(−1.14e3+1.88e3i)T2 1 + (-12.8 - 22.8i)T + (-1.14e3 + 1.88e3i)T^{2}
53 1+(−9.32−21.0i)T+(−1.88e3+2.07e3i)T2 1 + (-9.32 - 21.0i)T + (-1.88e3 + 2.07e3i)T^{2}
59 1+(60.4+7.79i)T+(3.36e3+882.i)T2 1 + (60.4 + 7.79i)T + (3.36e3 + 882. i)T^{2}
61 1+(4.06+6.70i)T+(−1.72e3+3.29e3i)T2 1 + (4.06 + 6.70i)T + (-1.72e3 + 3.29e3i)T^{2}
67 1+(−38.0−10.6i)T+(3.83e3+2.32e3i)T2 1 + (-38.0 - 10.6i)T + (3.83e3 + 2.32e3i)T^{2}
71 1+(34.9+10.9i)T+(4.13e3+2.88e3i)T2 1 + (34.9 + 10.9i)T + (4.13e3 + 2.88e3i)T^{2}
73 1+(33.6+29.5i)T+(681.+5.28e3i)T2 1 + (33.6 + 29.5i)T + (681. + 5.28e3i)T^{2}
79 1+(−9.25+63.7i)T+(−5.98e3−1.77e3i)T2 1 + (-9.25 + 63.7i)T + (-5.98e3 - 1.77e3i)T^{2}
83 1+(−24.6−19.6i)T+(1.53e3+6.71e3i)T2 1 + (-24.6 - 19.6i)T + (1.53e3 + 6.71e3i)T^{2}
89 1+(−55.7−67.6i)T+(−1.51e3+7.77e3i)T2 1 + (-55.7 - 67.6i)T + (-1.51e3 + 7.77e3i)T^{2}
97 1+(−9.92−37.8i)T+(−8.19e3+4.61e3i)T2 1 + (-9.92 - 37.8i)T + (-8.19e3 + 4.61e3i)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

−11.75185922683394518536736989327, −10.67796000847605185933104874093, −9.525763852231340130551574801297, −8.938178948699795023411157760943, −7.913733930292503413162390412781, −7.34552642661217432038368299151, −5.21726181063776218413809998886, −4.42312937803580897615095806133, −1.62996000947607112229369180198, −0.46305368374877214203514772884, 1.98518867814011711574755662911, 3.46375942404978758136864825325, 5.73289790247313947082321444029, 6.87467405928019205133674872836, 7.66850274690673118157576118306, 8.810716104780362523741008199417, 9.583877002847192271971100655812, 10.73235882748992247167608193418, 11.56957712675361999912912459169, 11.83515950567142780942179151131

Graph of the ZZ-function along the critical line