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Properties

Label 2-338-13.12-c7-0-20
Degree 22
Conductor 338338
Sign 0.554+0.832i0.554 + 0.832i
Analytic cond. 105.586105.586
Root an. cond. 10.275510.2755
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − 39.9·3-s − 64·4-s − 323. i·5-s + 319. i·6-s − 568. i·7-s + 512i·8-s − 588.·9-s − 2.58e3·10-s + 238. i·11-s + 2.55e3·12-s − 4.54e3·14-s + 1.29e4i·15-s + 4.09e3·16-s − 2.04e4·17-s + 4.70e3i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.855·3-s − 0.5·4-s − 1.15i·5-s + 0.604i·6-s − 0.626i·7-s + 0.353i·8-s − 0.268·9-s − 0.818·10-s + 0.0539i·11-s + 0.427·12-s − 0.443·14-s + 0.990i·15-s + 0.250·16-s − 1.01·17-s + 0.190i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.554+0.832i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+7/2) L(s)=((0.554+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.554+0.832i0.554 + 0.832i
Analytic conductor: 105.586105.586
Root analytic conductor: 10.275510.2755
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ338(337,⋅)\chi_{338} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :7/2), 0.554+0.832i)(2,\ 338,\ (\ :7/2),\ 0.554 + 0.832i)

Particular Values

L(4)L(4) ≈\approx 0.86196104090.8619610409
L(12)L(\frac12) ≈\approx 0.86196104090.8619610409
L(92)L(\frac{9}{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+8iT 1 + 8iT
13 1 1
good3 1+39.9T+2.18e3T2 1 + 39.9T + 2.18e3T^{2}
5 1+323.iT−7.81e4T2 1 + 323. iT - 7.81e4T^{2}
7 1+568.iT−8.23e5T2 1 + 568. iT - 8.23e5T^{2}
11 1−238.iT−1.94e7T2 1 - 238. iT - 1.94e7T^{2}
17 1+2.04e4T+4.10e8T2 1 + 2.04e4T + 4.10e8T^{2}
19 1+9.64e3iT−8.93e8T2 1 + 9.64e3iT - 8.93e8T^{2}
23 1−7.82e4T+3.40e9T2 1 - 7.82e4T + 3.40e9T^{2}
29 1+1.38e5T+1.72e10T2 1 + 1.38e5T + 1.72e10T^{2}
31 1−1.60e5iT−2.75e10T2 1 - 1.60e5iT - 2.75e10T^{2}
37 1−1.52e5iT−9.49e10T2 1 - 1.52e5iT - 9.49e10T^{2}
41 1−1.85e5iT−1.94e11T2 1 - 1.85e5iT - 1.94e11T^{2}
43 1+8.50e4T+2.71e11T2 1 + 8.50e4T + 2.71e11T^{2}
47 1−1.20e6iT−5.06e11T2 1 - 1.20e6iT - 5.06e11T^{2}
53 1+6.65e5T+1.17e12T2 1 + 6.65e5T + 1.17e12T^{2}
59 1−2.48e6iT−2.48e12T2 1 - 2.48e6iT - 2.48e12T^{2}
61 1+3.04e6T+3.14e12T2 1 + 3.04e6T + 3.14e12T^{2}
67 1+3.87e5iT−6.06e12T2 1 + 3.87e5iT - 6.06e12T^{2}
71 1−3.68e6iT−9.09e12T2 1 - 3.68e6iT - 9.09e12T^{2}
73 1+1.57e6iT−1.10e13T2 1 + 1.57e6iT - 1.10e13T^{2}
79 1−2.29e6T+1.92e13T2 1 - 2.29e6T + 1.92e13T^{2}
83 1+7.93e6iT−2.71e13T2 1 + 7.93e6iT - 2.71e13T^{2}
89 1+8.15e6iT−4.42e13T2 1 + 8.15e6iT - 4.42e13T^{2}
97 1+1.33e6iT−8.07e13T2 1 + 1.33e6iT - 8.07e13T^{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

−10.50909254446312383968265436834, −9.199215587392234387577152143507, −8.700732420312511108914972053852, −7.34892037427294304797794247441, −6.11833692512371083802136061119, −4.97212976653520019603767483783, −4.48330134204245843503117292351, −3.00923689381029549655105776521, −1.43293200281877317531790896554, −0.61387212433607916546038049633, 0.36353430297368156699168250972, 2.25677612196461747525294693063, 3.47110058126460420576348927063, 4.92440304809026562595712609931, 5.82787707316120928387479520525, 6.55327912886416284384544177417, 7.33837318659766354421136572878, 8.576096320038673397667656923542, 9.471158897555456492744624305880, 10.76771826858610253369946921657

Graph of the ZZ-function along the critical line