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Properties

Label 2-639-639.70-c0-0-1
Degree $2$
Conductor $639$
Sign $-0.124 - 0.992i$
Analytic cond. $0.318902$
Root an. cond. $0.564714$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.955 + 1.65i)2-s + (0.365 − 0.930i)3-s + (−1.32 − 2.29i)4-s + (0.900 + 1.56i)5-s + (1.19 + 1.49i)6-s + 3.15·8-s + (−0.733 − 0.680i)9-s − 3.44·10-s + (−2.62 + 0.395i)12-s + (1.78 − 0.268i)15-s + (−1.69 + 2.92i)16-s + (1.82 − 0.563i)18-s + 1.91·19-s + (2.38 − 4.13i)20-s + (1.15 − 2.93i)24-s + (−1.12 + 1.94i)25-s + ⋯
L(s)  = 1  + (−0.955 + 1.65i)2-s + (0.365 − 0.930i)3-s + (−1.32 − 2.29i)4-s + (0.900 + 1.56i)5-s + (1.19 + 1.49i)6-s + 3.15·8-s + (−0.733 − 0.680i)9-s − 3.44·10-s + (−2.62 + 0.395i)12-s + (1.78 − 0.268i)15-s + (−1.69 + 2.92i)16-s + (1.82 − 0.563i)18-s + 1.91·19-s + (2.38 − 4.13i)20-s + (1.15 − 2.93i)24-s + (−1.12 + 1.94i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(0.318902\)
Root analytic conductor: \(0.564714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :0),\ -0.124 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6970096447\)
\(L(\frac12)\) \(\approx\) \(0.6970096447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.365 + 0.930i)T \)
71 \( 1 - T \)
good2 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.91T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.445T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.80T + T^{2} \)
79 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 1.97T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61785330423784994575228819683, −9.770460922616687562309011066412, −9.199132592654245094103864162009, −8.109585770049517219759049908973, −7.22887820308720721931643357279, −6.93736683513086996013612873335, −6.02042850480146554968428877591, −5.37850990696452514818044680566, −3.16062017525899134243488388278, −1.63923051076243324667011978505, 1.27675691423941715251215364818, 2.50123260754730766957914442511, 3.68360468689696517314257436523, 4.69941678121741007453852726438, 5.48990115714449624891753291548, 7.74764998127215504086105849237, 8.494028070058355056396437597299, 9.276075807704295241712224687248, 9.601499748929168905444309000684, 10.24925108108803823380641352491

Graph of the $Z$-function along the critical line