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Properties

Label 2-693-231.167-c0-0-1
Degree 22
Conductor 693693
Sign −0.147−0.989i-0.147 - 0.989i
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  + (−0.610 + 1.87i)2-s + (−2.34 − 1.70i)4-s + (0.587 − 0.809i)7-s + (3.03 − 2.20i)8-s + (0.891 + 0.453i)11-s + (1.16 + 1.59i)14-s + (1.39 + 4.29i)16-s + (−1.39 + 1.39i)22-s + 0.312i·23-s + (0.809 − 0.587i)25-s + (−2.76 + 0.896i)28-s + (−0.734 − 0.533i)29-s − 5.17·32-s + (0.951 + 0.690i)37-s + 0.618i·43-s + (−1.31 − 2.58i)44-s + ⋯
L(s)  = 1  + (−0.610 + 1.87i)2-s + (−2.34 − 1.70i)4-s + (0.587 − 0.809i)7-s + (3.03 − 2.20i)8-s + (0.891 + 0.453i)11-s + (1.16 + 1.59i)14-s + (1.39 + 4.29i)16-s + (−1.39 + 1.39i)22-s + 0.312i·23-s + (0.809 − 0.587i)25-s + (−2.76 + 0.896i)28-s + (−0.734 − 0.533i)29-s − 5.17·32-s + (0.951 + 0.690i)37-s + 0.618i·43-s + (−1.31 − 2.58i)44-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=((−0.147−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s) L(s)=((−0.147−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: −0.147−0.989i-0.147 - 0.989i
Analytic conductor: 0.3458520.345852
Root analytic conductor: 0.5880910.588091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(629,⋅)\chi_{693} (629, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 693, ( :0), −0.147−0.989i)(2,\ 693,\ (\ :0),\ -0.147 - 0.989i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.67724969110.6772496911
L(12)L(\frac12) ≈\approx 0.67724969110.6772496911
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+(−0.891−0.453i)T 1 + (-0.891 - 0.453i)T
good2 1+(0.610−1.87i)T+(−0.809−0.587i)T2 1 + (0.610 - 1.87i)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 1−0.312iT−T2 1 - 0.312iT - T^{2}
29 1+(0.734+0.533i)T+(0.309+0.951i)T2 1 + (0.734 + 0.533i)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+(−0.863−0.280i)T+(0.809+0.587i)T2 1 + (-0.863 - 0.280i)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+(1.69−0.550i)T+(0.809−0.587i)T2 1 + (1.69 - 0.550i)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

−10.45936253091187951372935840207, −9.757995265342043893447273571597, −8.948127302568418380120890984609, −8.122350830311265845999788676471, −7.37085698485431666545576085081, −6.71816112347305667068705749878, −5.82375140160923125137920304210, −4.71336327498026176359474484415, −4.07489034985111058597120614432, −1.26000879937970853575463934527, 1.35212972536604054835604403879, 2.47461377835267084633176270288, 3.53562106782954021963187066022, 4.53278487157347347989921234145, 5.60046733470961286680597776639, 7.31956426218587169612644952584, 8.436389323149689825361390184033, 8.922204059075404120814799321401, 9.592240195721238153529112208441, 10.64726863279264593003175835788

Graph of the ZZ-function along the critical line