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Properties

Label 2-14e2-7.2-c5-0-5
Degree 22
Conductor 196196
Sign −0.386−0.922i-0.386 - 0.922i
Analytic cond. 31.435231.4352
Root an. cond. 5.606715.60671
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8 + 13.8i)3-s + (−8 + 13.8i)5-s + (−6.49 + 11.2i)9-s + (38 + 65.8i)11-s + 880·13-s − 255.·15-s + (528 + 914. i)17-s + (−968 + 1.67e3i)19-s + (−468 + 810. i)23-s + (1.43e3 + 2.48e3i)25-s + 3.68e3·27-s − 3.98e3·29-s + (−784 − 1.35e3i)31-s + (−607. + 1.05e3i)33-s + (−2.46e3 + 4.27e3i)37-s + ⋯
L(s)  = 1  + (0.513 + 0.888i)3-s + (−0.143 + 0.247i)5-s + (−0.0267 + 0.0463i)9-s + (0.0946 + 0.164i)11-s + 1.44·13-s − 0.293·15-s + (0.443 + 0.767i)17-s + (−0.615 + 1.06i)19-s + (−0.184 + 0.319i)23-s + (0.459 + 0.795i)25-s + 0.971·27-s − 0.879·29-s + (−0.146 − 0.253i)31-s + (−0.0971 + 0.168i)33-s + (−0.296 + 0.513i)37-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=((−0.386−0.922i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=((−0.386−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −0.386−0.922i-0.386 - 0.922i
Analytic conductor: 31.435231.4352
Root analytic conductor: 5.606715.60671
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ196(177,⋅)\chi_{196} (177, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 196, ( :5/2), −0.386−0.922i)(2,\ 196,\ (\ :5/2),\ -0.386 - 0.922i)

Particular Values

L(3)L(3) ≈\approx 2.2233254162.223325416
L(12)L(\frac12) ≈\approx 2.2233254162.223325416
L(72)L(\frac{7}{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
7 1 1
good3 1+(−8−13.8i)T+(−121.5+210.i)T2 1 + (-8 - 13.8i)T + (-121.5 + 210. i)T^{2}
5 1+(8−13.8i)T+(−1.56e3−2.70e3i)T2 1 + (8 - 13.8i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−38−65.8i)T+(−8.05e4+1.39e5i)T2 1 + (-38 - 65.8i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−880T+3.71e5T2 1 - 880T + 3.71e5T^{2}
17 1+(−528−914.i)T+(−7.09e5+1.22e6i)T2 1 + (-528 - 914. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(968−1.67e3i)T+(−1.23e6−2.14e6i)T2 1 + (968 - 1.67e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(468−810.i)T+(−3.21e6−5.57e6i)T2 1 + (468 - 810. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1+3.98e3T+2.05e7T2 1 + 3.98e3T + 2.05e7T^{2}
31 1+(784+1.35e3i)T+(−1.43e7+2.47e7i)T2 1 + (784 + 1.35e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(2.46e3−4.27e3i)T+(−3.46e7−6.00e7i)T2 1 + (2.46e3 - 4.27e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+1.58e4T+1.15e8T2 1 + 1.58e4T + 1.15e8T^{2}
43 1+1.64e4T+1.47e8T2 1 + 1.64e4T + 1.47e8T^{2}
47 1+(−1.03e4+1.79e4i)T+(−1.14e8−1.98e8i)T2 1 + (-1.03e4 + 1.79e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−1.87e4−3.23e4i)T+(−2.09e8+3.62e8i)T2 1 + (-1.87e4 - 3.23e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.05e4+1.83e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.05e4 + 1.83e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−1.49e3+2.59e3i)T+(−4.22e8−7.31e8i)T2 1 + (-1.49e3 + 2.59e3i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−2.29e4−3.96e4i)T+(−6.75e8+1.16e9i)T2 1 + (-2.29e4 - 3.96e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+4.98e4T+1.80e9T2 1 + 4.98e4T + 1.80e9T^{2}
73 1+(−2.81e4−4.87e4i)T+(−1.03e9+1.79e9i)T2 1 + (-2.81e4 - 4.87e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(2.03e4−3.52e4i)T+(−1.53e9−2.66e9i)T2 1 + (2.03e4 - 3.52e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−1.12e5T+3.93e9T2 1 - 1.12e5T + 3.93e9T^{2}
89 1+(3.21e4−5.56e4i)T+(−2.79e9−4.83e9i)T2 1 + (3.21e4 - 5.56e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+2.27e3T+8.58e9T2 1 + 2.27e3T + 8.58e9T^{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

−11.81561721452964933516469377766, −10.69561159184162142957522118339, −10.03886602568073940224929296641, −8.913900183961893331339833135641, −8.169400650125239181379762570001, −6.73325904190581035123164386360, −5.55214891417132274776228048930, −3.97068208298486547358754977853, −3.43351682167006167427289163902, −1.53492376867214420360106214468, 0.66966273850264665563748043896, 1.95317081922379601423345697329, 3.34176579791895165297360408445, 4.83078896996562077526495690341, 6.30201142062489954899870952425, 7.23631401458945937679305316037, 8.337777881192578572905154500694, 8.942898202073100229609197807777, 10.41391690129658148311588028599, 11.40397241812851182311984451600

Graph of the ZZ-function along the critical line