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Properties

Label 2-329-329.93-c0-0-1
Degree 22
Conductor 329329
Sign −0.784−0.620i-0.784 - 0.620i
Analytic cond. 0.1641920.164192
Root an. cond. 0.4052060.405206
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  + (−0.669 + 1.15i)2-s + (0.809 + 1.40i)3-s + (−0.395 − 0.684i)4-s − 2.16·6-s + (0.669 − 0.743i)7-s − 0.279·8-s + (−0.809 + 1.40i)9-s + (0.639 − 1.10i)12-s + (0.413 + 1.27i)14-s + (0.582 − 1.00i)16-s + (−0.913 − 1.58i)17-s + (−1.08 − 1.87i)18-s + (1.58 + 0.336i)21-s + (−0.226 − 0.392i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.669 + 1.15i)2-s + (0.809 + 1.40i)3-s + (−0.395 − 0.684i)4-s − 2.16·6-s + (0.669 − 0.743i)7-s − 0.279·8-s + (−0.809 + 1.40i)9-s + (0.639 − 1.10i)12-s + (0.413 + 1.27i)14-s + (0.582 − 1.00i)16-s + (−0.913 − 1.58i)17-s + (−1.08 − 1.87i)18-s + (1.58 + 0.336i)21-s + (−0.226 − 0.392i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

Λ(s)=(329s/2 ΓC(s) L(s)=((−0.784−0.620i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 329 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(329s/2 ΓC(s) L(s)=((−0.784−0.620i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 329 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 329329    =    7⋅477 \cdot 47
Sign: −0.784−0.620i-0.784 - 0.620i
Analytic conductor: 0.1641920.164192
Root analytic conductor: 0.4052060.405206
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ329(93,⋅)\chi_{329} (93, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 329, ( :0), −0.784−0.620i)(2,\ 329,\ (\ :0),\ -0.784 - 0.620i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73338557000.7333855700
L(12)L(\frac12) ≈\approx 0.73338557000.7333855700
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)
bad7 1+(−0.669+0.743i)T 1 + (-0.669 + 0.743i)T
47 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(0.669−1.15i)T+(−0.5−0.866i)T2 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2}
3 1+(−0.809−1.40i)T+(−0.5+0.866i)T2 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.913+1.58i)T+(−0.5+0.866i)T2 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.913−1.58i)T+(−0.5−0.866i)T2 1 + (0.913 - 1.58i)T + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
53 1+(−0.978−1.69i)T+(−0.5+0.866i)T2 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2}
59 1+(0.669+1.15i)T+(−0.5+0.866i)T2 1 + (0.669 + 1.15i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.669−1.15i)T+(−0.5−0.866i)T2 1 + (0.669 - 1.15i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−1.82T+T2 1 - 1.82T + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−0.809+1.40i)T+(−0.5−0.866i)T2 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2}
83 1+T+T2 1 + T + T^{2}
89 1+(0.309−0.535i)T+(−0.5−0.866i)T2 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2}
97 1+1.95T+T2 1 + 1.95T + 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

−11.91438539982107099670360261650, −10.87196333030955357408634184581, −9.948483395748035623363736136830, −9.236255472473650207245477498343, −8.449914372338562942614672988005, −7.67625808944133744472150545490, −6.65558505077137405488207586256, −5.13171704548076107549426400459, −4.31284991009843045661816457700, −2.90183104402826162231304474382, 1.68645756168233095062941011887, 2.26449363185006760463762289466, 3.64692453493006510428542000412, 5.71329851781330750415844131549, 6.85407118795595001850969981394, 8.125258701577221978287428657344, 8.588071381174095460277208669792, 9.409601448047303321602950053638, 10.72225150387065585578446362631, 11.47569131598313690268391917731

Graph of the ZZ-function along the critical line