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Properties

Label 2-13e2-13.12-c1-0-3
Degree 22
Conductor 169169
Sign 0.277−0.960i0.277 - 0.960i
Analytic cond. 1.349471.34947
Root an. cond. 1.161661.16166
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s + 2·3-s − 0.999·4-s − 1.73i·5-s + 3.46i·6-s + 1.73i·8-s + 9-s + 2.99·10-s − 1.99·12-s − 3.46i·15-s − 5·16-s − 3·17-s + 1.73i·18-s − 3.46i·19-s + 1.73i·20-s + ⋯
L(s)  = 1  + 1.22i·2-s + 1.15·3-s − 0.499·4-s − 0.774i·5-s + 1.41i·6-s + 0.612i·8-s + 0.333·9-s + 0.948·10-s − 0.577·12-s − 0.894i·15-s − 1.25·16-s − 0.727·17-s + 0.408i·18-s − 0.794i·19-s + 0.387i·20-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.277−0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.277−0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.277−0.960i0.277 - 0.960i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(168,⋅)\chi_{169} (168, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), 0.277−0.960i)(2,\ 169,\ (\ :1/2),\ 0.277 - 0.960i)

Particular Values

L(1)L(1) ≈\approx 1.26684+0.952866i1.26684 + 0.952866i
L(12)L(\frac12) ≈\approx 1.26684+0.952866i1.26684 + 0.952866i
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)
bad13 1 1
good2 1−1.73iT−2T2 1 - 1.73iT - 2T^{2}
3 1−2T+3T2 1 - 2T + 3T^{2}
5 1+1.73iT−5T2 1 + 1.73iT - 5T^{2}
7 1−7T2 1 - 7T^{2}
11 1−11T2 1 - 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1+3.46iT−19T2 1 + 3.46iT - 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1+8.66iT−37T2 1 + 8.66iT - 37T^{2}
41 1−5.19iT−41T2 1 - 5.19iT - 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1+3.46iT−47T2 1 + 3.46iT - 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1−6.92iT−59T2 1 - 6.92iT - 59T^{2}
61 1−T+61T2 1 - T + 61T^{2}
67 1+3.46iT−67T2 1 + 3.46iT - 67T^{2}
71 1−3.46iT−71T2 1 - 3.46iT - 71T^{2}
73 1−1.73iT−73T2 1 - 1.73iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−13.8iT−83T2 1 - 13.8iT - 83T^{2}
89 1−6.92iT−89T2 1 - 6.92iT - 89T^{2}
97 1−6.92iT−97T2 1 - 6.92iT - 97T^{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

−13.44476425929559468346538491569, −12.26818101199900444750312981580, −10.94847564488489828881242774172, −9.332238110505233578204456023620, −8.673371113803740723619786588554, −7.939310022246744940027896702216, −6.86820121374391796497923239343, −5.57603028151523584184682147069, −4.30239335282076746516779963742, −2.44501112123847370636860148022, 2.10572921794485249831794009614, 3.06189891578492420096607173424, 4.11034874790725181817896976209, 6.30914840314477631398284782079, 7.58992859012226347336486584256, 8.735107569485189719239757586320, 9.780212139297250293402784529936, 10.56403233314047195907411460948, 11.52441054296548637032148096270, 12.48263861716559487451818949027

Graph of the ZZ-function along the critical line