↑

Properties

Label 2-14e2-196.111-c1-0-20
Degree 22
Conductor 196196
Sign 0.955+0.295i0.955 + 0.295i
Analytic cond. 1.565061.56506
Root an. cond. 1.251021.25102
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.04 − 0.957i)2-s + (1.57 + 1.96i)3-s + (0.168 − 1.99i)4-s + (1.55 − 1.23i)5-s + (3.52 + 0.547i)6-s + (−2.03 + 1.69i)7-s + (−1.73 − 2.23i)8-s + (−0.745 + 3.26i)9-s + (0.432 − 2.77i)10-s + (−3.09 + 0.707i)11-s + (4.18 − 2.79i)12-s + (−5.38 + 1.22i)13-s + (−0.498 + 3.70i)14-s + (4.88 + 1.11i)15-s + (−3.94 − 0.670i)16-s + (2.48 − 5.15i)17-s + ⋯
L(s)  = 1  + (0.736 − 0.676i)2-s + (0.906 + 1.13i)3-s + (0.0840 − 0.996i)4-s + (0.695 − 0.554i)5-s + (1.43 + 0.223i)6-s + (−0.768 + 0.639i)7-s + (−0.612 − 0.790i)8-s + (−0.248 + 1.08i)9-s + (0.136 − 0.878i)10-s + (−0.934 + 0.213i)11-s + (1.20 − 0.808i)12-s + (−1.49 + 0.340i)13-s + (−0.133 + 0.991i)14-s + (1.26 + 0.287i)15-s + (−0.985 − 0.167i)16-s + (0.601 − 1.24i)17-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((0.955+0.295i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+1/2) L(s)=((0.955+0.295i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: 0.955+0.295i0.955 + 0.295i
Analytic conductor: 1.565061.56506
Root analytic conductor: 1.251021.25102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ196(111,⋅)\chi_{196} (111, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :1/2), 0.955+0.295i)(2,\ 196,\ (\ :1/2),\ 0.955 + 0.295i)

Particular Values

L(1)L(1) ≈\approx 2.08034−0.313948i2.08034 - 0.313948i
L(12)L(\frac12) ≈\approx 2.08034−0.313948i2.08034 - 0.313948i
L(32)L(\frac{3}{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+(−1.04+0.957i)T 1 + (-1.04 + 0.957i)T
7 1+(2.03−1.69i)T 1 + (2.03 - 1.69i)T
good3 1+(−1.57−1.96i)T+(−0.667+2.92i)T2 1 + (-1.57 - 1.96i)T + (-0.667 + 2.92i)T^{2}
5 1+(−1.55+1.23i)T+(1.11−4.87i)T2 1 + (-1.55 + 1.23i)T + (1.11 - 4.87i)T^{2}
11 1+(3.09−0.707i)T+(9.91−4.77i)T2 1 + (3.09 - 0.707i)T + (9.91 - 4.77i)T^{2}
13 1+(5.38−1.22i)T+(11.7−5.64i)T2 1 + (5.38 - 1.22i)T + (11.7 - 5.64i)T^{2}
17 1+(−2.48+5.15i)T+(−10.5−13.2i)T2 1 + (-2.48 + 5.15i)T + (-10.5 - 13.2i)T^{2}
19 1−7.59T+19T2 1 - 7.59T + 19T^{2}
23 1+(−0.751−1.55i)T+(−14.3+17.9i)T2 1 + (-0.751 - 1.55i)T + (-14.3 + 17.9i)T^{2}
29 1+(−0.626−0.301i)T+(18.0+22.6i)T2 1 + (-0.626 - 0.301i)T + (18.0 + 22.6i)T^{2}
31 1+8.05T+31T2 1 + 8.05T + 31T^{2}
37 1+(−4.24−2.04i)T+(23.0+28.9i)T2 1 + (-4.24 - 2.04i)T + (23.0 + 28.9i)T^{2}
41 1+(0.0970−0.0773i)T+(9.12−39.9i)T2 1 + (0.0970 - 0.0773i)T + (9.12 - 39.9i)T^{2}
43 1+(3.53+2.81i)T+(9.56+41.9i)T2 1 + (3.53 + 2.81i)T + (9.56 + 41.9i)T^{2}
47 1+(−0.299−1.31i)T+(−42.3+20.3i)T2 1 + (-0.299 - 1.31i)T + (-42.3 + 20.3i)T^{2}
53 1+(−0.183+0.0884i)T+(33.0−41.4i)T2 1 + (-0.183 + 0.0884i)T + (33.0 - 41.4i)T^{2}
59 1+(−1.40+1.76i)T+(−13.1−57.5i)T2 1 + (-1.40 + 1.76i)T + (-13.1 - 57.5i)T^{2}
61 1+(−0.524+1.08i)T+(−38.0−47.6i)T2 1 + (-0.524 + 1.08i)T + (-38.0 - 47.6i)T^{2}
67 1−3.71iT−67T2 1 - 3.71iT - 67T^{2}
71 1+(−3.42−7.11i)T+(−44.2+55.5i)T2 1 + (-3.42 - 7.11i)T + (-44.2 + 55.5i)T^{2}
73 1+(−4.61−1.05i)T+(65.7+31.6i)T2 1 + (-4.61 - 1.05i)T + (65.7 + 31.6i)T^{2}
79 1+9.99iT−79T2 1 + 9.99iT - 79T^{2}
83 1+(0.712−3.12i)T+(−74.7−36.0i)T2 1 + (0.712 - 3.12i)T + (-74.7 - 36.0i)T^{2}
89 1+(−14.8−3.38i)T+(80.1+38.6i)T2 1 + (-14.8 - 3.38i)T + (80.1 + 38.6i)T^{2}
97 1+3.30iT−97T2 1 + 3.30iT - 97T^{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

−12.55440896627499980084445453130, −11.61938895117308715832859662879, −10.07777061883088520295446107280, −9.603468683642423225285906882156, −9.196425649424016647575140029358, −7.33898344789991881400230652559, −5.39030496015750653749336225199, −5.00068880229603840637837373321, −3.32786904440539284213164332162, −2.47853955719616050727838115526, 2.46337090368210571077251427719, 3.36711194796252915514281565424, 5.36499230183182803013202877550, 6.50298114340818633096466249985, 7.46100841187498582417516901444, 7.899389600321459782877685197492, 9.439752795377984949325272864340, 10.49881660107492147250323224062, 12.20314931366054305651439906705, 12.92287510094218929709998181684

Graph of the ZZ-function along the critical line