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Properties

Label 2-693-231.83-c0-0-3
Degree 22
Conductor 693693
Sign 0.190+0.981i0.190 + 0.981i
Analytic cond. 0.3458520.345852
Root an. cond. 0.5880910.588091
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  + (−0.0966 − 0.297i)2-s + (0.729 − 0.530i)4-s + (−0.587 − 0.809i)7-s + (−0.481 − 0.349i)8-s + (−0.453 − 0.891i)11-s + (−0.183 + 0.253i)14-s + (0.221 − 0.680i)16-s + (−0.221 + 0.221i)22-s + 1.97i·23-s + (0.809 + 0.587i)25-s + (−0.857 − 0.278i)28-s + (1.44 − 1.04i)29-s − 0.819·32-s + (−0.951 + 0.690i)37-s + 0.618i·43-s + (−0.803 − 0.409i)44-s + ⋯
L(s)  = 1  + (−0.0966 − 0.297i)2-s + (0.729 − 0.530i)4-s + (−0.587 − 0.809i)7-s + (−0.481 − 0.349i)8-s + (−0.453 − 0.891i)11-s + (−0.183 + 0.253i)14-s + (0.221 − 0.680i)16-s + (−0.221 + 0.221i)22-s + 1.97i·23-s + (0.809 + 0.587i)25-s + (−0.857 − 0.278i)28-s + (1.44 − 1.04i)29-s − 0.819·32-s + (−0.951 + 0.690i)37-s + 0.618i·43-s + (−0.803 − 0.409i)44-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((0.190+0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s) L(s)=((0.190+0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 0.190+0.981i0.190 + 0.981i
Analytic conductor: 0.3458520.345852
Root analytic conductor: 0.5880910.588091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(314,⋅)\chi_{693} (314, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :0), 0.190+0.981i)(2,\ 693,\ (\ :0),\ 0.190 + 0.981i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.95951921210.9595192121
L(12)L(\frac12) ≈\approx 0.95951921210.9595192121
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
7 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
11 1+(0.453+0.891i)T 1 + (0.453 + 0.891i)T
good2 1+(0.0966+0.297i)T+(−0.809+0.587i)T2 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2}
5 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
13 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
17 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
19 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
23 1−1.97iT−T2 1 - 1.97iT - T^{2}
29 1+(−1.44+1.04i)T+(0.309−0.951i)T2 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2}
31 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(0.951−0.690i)T+(0.309−0.951i)T2 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2}
41 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
43 1−0.618iT−T2 1 - 0.618iT - T^{2}
47 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
53 1+(−1.69+0.550i)T+(0.809−0.587i)T2 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2}
59 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
61 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
67 1−0.618T+T2 1 - 0.618T + T^{2}
71 1+(0.863+0.280i)T+(0.809+0.587i)T2 1 + (0.863 + 0.280i)T + (0.809 + 0.587i)T^{2}
73 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
79 1+(1.80−0.587i)T+(0.809−0.587i)T2 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2}
83 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)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

−10.37594545041502237276629913250, −9.951886311050152722803010337681, −8.910670978511422176849034732771, −7.74804915904844083266723294502, −6.92490468638593305273735200778, −6.09563358329533034294729899154, −5.16992503337588871658804787849, −3.65045408247538775816267399231, −2.79757146720682771443586220879, −1.16817165119937264475905190244, 2.27602869291818794927641599057, 3.01177590942118824597178671337, 4.49216267179879403719268958528, 5.64094936024933737088868088790, 6.67387597092873804679454244932, 7.13398731649512979840997806798, 8.453802646447462222938220823839, 8.792458705505487334432440055939, 10.16942601512139081162810224803, 10.68151005833140564249556600798

Graph of the ZZ-function along the critical line