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Properties

Label 2-1305-435.434-c0-0-3
Degree 22
Conductor 13051305
Sign 0.169−0.985i0.169 - 0.985i
Analytic cond. 0.6512790.651279
Root an. cond. 0.8070190.807019
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.707 + 0.707i)5-s − i·7-s + i·8-s + (−0.707 + 0.707i)10-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + 1.41i·19-s + i·22-s + 1.00i·25-s + 26-s − 29-s + ⋯
L(s)  = 1  + i·2-s + (0.707 + 0.707i)5-s − i·7-s + i·8-s + (−0.707 + 0.707i)10-s + 11-s − i·13-s + 14-s − 16-s − i·17-s + 1.41i·19-s + i·22-s + 1.00i·25-s + 26-s − 29-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=((0.169−0.985i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s) L(s)=((0.169−0.985i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 0.169−0.985i0.169 - 0.985i
Analytic conductor: 0.6512790.651279
Root analytic conductor: 0.8070190.807019
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1305(1304,⋅)\chi_{1305} (1304, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1305, ( :0), 0.169−0.985i)(2,\ 1305,\ (\ :0),\ 0.169 - 0.985i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3841294821.384129482
L(12)L(\frac12) ≈\approx 1.3841294821.384129482
L(1)L(1) 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)
bad3 1 1
5 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
29 1+T 1 + T
good2 1−iT−T2 1 - iT - T^{2}
7 1+iT−T2 1 + iT - T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1−1.41iT−T2 1 - 1.41iT - T^{2}
23 1+T2 1 + T^{2}
31 1+1.41iT−T2 1 + 1.41iT - T^{2}
37 1+1.41T+T2 1 + 1.41T + T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1−iT−T2 1 - iT - T^{2}
53 1+1.41T+T2 1 + 1.41T + T^{2}
59 1−1.41iT−T2 1 - 1.41iT - T^{2}
61 1+1.41iT−T2 1 + 1.41iT - T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+T+T2 1 + T + T^{2}
97 1−1.41T+T2 1 - 1.41T + T^{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

−9.951341626290097909224608454913, −9.240714301375710485092467306224, −8.036741708688563176233231622346, −7.43599191633021730782861685397, −6.79702715156859495192709331649, −5.99960822400291924195681016217, −5.38250658370185532485174349822, −4.05344393099094602045311876221, −3.00698395384819701814263092080, −1.67038125642667276524652896145, 1.53299647145158683911279144529, 2.12252140003167944483305309268, 3.35522843904284125642139184055, 4.39497443868652757035883322767, 5.37591952760100189981128218137, 6.41102835994116180479947010341, 6.93685517473482952822409025129, 8.536126068260713750722347587293, 9.098126507622570628551320832310, 9.534675365412083283193699606776

Graph of the ZZ-function along the critical line