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Properties

Label 2-639-71.45-c1-0-21
Degree $2$
Conductor $639$
Sign $-0.649 + 0.760i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.154 − 0.0744i)2-s + (−1.22 + 1.54i)4-s − 0.566·5-s + (−1.38 − 0.668i)7-s + (−0.151 + 0.664i)8-s + (−0.0876 + 0.0421i)10-s + (−0.879 − 3.85i)11-s + (−0.834 + 3.65i)13-s − 0.264·14-s + (−0.850 − 3.72i)16-s + 4.10·17-s + (−2.81 − 3.53i)19-s + (0.696 − 0.873i)20-s + (−0.422 − 0.530i)22-s + (−7.97 − 3.83i)23-s + ⋯
L(s)  = 1  + (0.109 − 0.0526i)2-s + (−0.614 + 0.770i)4-s − 0.253·5-s + (−0.524 − 0.252i)7-s + (−0.0536 + 0.234i)8-s + (−0.0277 + 0.0133i)10-s + (−0.265 − 1.16i)11-s + (−0.231 + 1.01i)13-s − 0.0706·14-s + (−0.212 − 0.932i)16-s + 0.994·17-s + (−0.646 − 0.810i)19-s + (0.155 − 0.195i)20-s + (−0.0901 − 0.113i)22-s + (−1.66 − 0.800i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149075 - 0.323556i\)
\(L(\frac12)\) \(\approx\) \(0.149075 - 0.323556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + (-1.69 + 8.25i)T \)
good2 \( 1 + (-0.154 + 0.0744i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + 0.566T + 5T^{2} \)
7 \( 1 + (1.38 + 0.668i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (0.879 + 3.85i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.834 - 3.65i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 + (2.81 + 3.53i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (7.97 + 3.83i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.0365 + 0.0458i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-2.08 + 9.12i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-0.457 + 0.220i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (1.72 - 7.56i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-3.32 + 1.59i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (3.55 + 4.46i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.685 + 0.859i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-0.774 - 3.39i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-1.38 + 0.668i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (2.63 - 3.30i)T + (-14.9 - 65.3i)T^{2} \)
73 \( 1 + (11.4 - 5.49i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (3.47 - 15.2i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (-2.87 - 12.5i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-0.321 - 0.403i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-1.43 - 6.28i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.09958853014304276768447477782, −9.414422533506274563240067284488, −8.355212375693004059555259909784, −7.87906479107893986558392642598, −6.69017538251401520354271565869, −5.72532232672053128769832628513, −4.37525669381874417037110697824, −3.72503211382564591917563609067, −2.54309456463266207355859842705, −0.18379103936424101108744028408, 1.74999036577992164517833798935, 3.39132970430088303091278461368, 4.46257559475134761505104434131, 5.51409870817271103158062422106, 6.15276666072366611790994760924, 7.47749950412178145075879619102, 8.211703101130008558592555255958, 9.388160560931638790518407127853, 10.12375427784725844242517492880, 10.43120814832495951270808421376

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