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Properties

Label 2-13e2-13.12-c7-0-49
Degree 22
Conductor 169169
Sign 0.999+0.0304i0.999 + 0.0304i
Analytic cond. 52.793052.7930
Root an. cond. 7.265887.26588
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  + 3.19i·2-s − 57.0·3-s + 117.·4-s + 431. i·5-s − 182. i·6-s − 225. i·7-s + 785. i·8-s + 1.07e3·9-s − 1.37e3·10-s − 4.16e3i·11-s − 6.72e3·12-s + 719.·14-s − 2.46e4i·15-s + 1.25e4·16-s + 2.58e4·17-s + 3.42e3i·18-s + ⋯
L(s)  = 1  + 0.282i·2-s − 1.22·3-s + 0.920·4-s + 1.54i·5-s − 0.344i·6-s − 0.248i·7-s + 0.542i·8-s + 0.489·9-s − 0.436·10-s − 0.944i·11-s − 1.12·12-s + 0.0700·14-s − 1.88i·15-s + 0.766·16-s + 1.27·17-s + 0.138i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.999+0.0304i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((0.999+0.0304i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.999+0.0304i0.999 + 0.0304i
Analytic conductor: 52.793052.7930
Root analytic conductor: 7.265887.26588
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ169(168,⋅)\chi_{169} (168, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :7/2), 0.999+0.0304i)(2,\ 169,\ (\ :7/2),\ 0.999 + 0.0304i)

Particular Values

L(4)L(4) ≈\approx 1.4265668531.426566853
L(12)L(\frac12) ≈\approx 1.4265668531.426566853
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)
bad13 1 1
good2 1−3.19iT−128T2 1 - 3.19iT - 128T^{2}
3 1+57.0T+2.18e3T2 1 + 57.0T + 2.18e3T^{2}
5 1−431.iT−7.81e4T2 1 - 431. iT - 7.81e4T^{2}
7 1+225.iT−8.23e5T2 1 + 225. iT - 8.23e5T^{2}
11 1+4.16e3iT−1.94e7T2 1 + 4.16e3iT - 1.94e7T^{2}
17 1−2.58e4T+4.10e8T2 1 - 2.58e4T + 4.10e8T^{2}
19 1+4.20e4iT−8.93e8T2 1 + 4.20e4iT - 8.93e8T^{2}
23 1+7.24e4T+3.40e9T2 1 + 7.24e4T + 3.40e9T^{2}
29 1+1.74e5T+1.72e10T2 1 + 1.74e5T + 1.72e10T^{2}
31 1+2.90e5iT−2.75e10T2 1 + 2.90e5iT - 2.75e10T^{2}
37 1+1.32e5iT−9.49e10T2 1 + 1.32e5iT - 9.49e10T^{2}
41 1−6.40e5iT−1.94e11T2 1 - 6.40e5iT - 1.94e11T^{2}
43 1+1.75e5T+2.71e11T2 1 + 1.75e5T + 2.71e11T^{2}
47 1+4.13e5iT−5.06e11T2 1 + 4.13e5iT - 5.06e11T^{2}
53 1−1.28e6T+1.17e12T2 1 - 1.28e6T + 1.17e12T^{2}
59 1+1.35e6iT−2.48e12T2 1 + 1.35e6iT - 2.48e12T^{2}
61 1−2.73e6T+3.14e12T2 1 - 2.73e6T + 3.14e12T^{2}
67 1−7.89e5iT−6.06e12T2 1 - 7.89e5iT - 6.06e12T^{2}
71 1+2.07e6iT−9.09e12T2 1 + 2.07e6iT - 9.09e12T^{2}
73 1+3.16e6iT−1.10e13T2 1 + 3.16e6iT - 1.10e13T^{2}
79 1−6.20e6T+1.92e13T2 1 - 6.20e6T + 1.92e13T^{2}
83 1−1.25e5iT−2.71e13T2 1 - 1.25e5iT - 2.71e13T^{2}
89 1−1.96e6iT−4.42e13T2 1 - 1.96e6iT - 4.42e13T^{2}
97 1−5.34e6iT−8.07e13T2 1 - 5.34e6iT - 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

−11.30457015638466698254214033712, −10.84402428554196987390758000931, −9.909351251970794812988568033451, −7.915379612975953928392229631035, −7.02249667585445518125117615583, −6.17912600943939817939897248388, −5.55727130960444854782397695844, −3.54307890267009814905692846216, −2.37713474308323682708060897668, −0.52363651255751518993174780611, 0.955815508546847644111075177259, 1.85047565975963797455098176828, 3.88079106070405301494446063228, 5.31965227032585903708635270997, 5.81008158071309551458300330669, 7.21776066762593758185422572851, 8.363552238043402513572302325611, 9.797797498455614344160187500529, 10.52009605642865559828452382928, 12.00638075854582445923584832832

Graph of the ZZ-function along the critical line