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Properties

Label 2-639-71.32-c1-0-21
Degree $2$
Conductor $639$
Sign $0.808 - 0.588i$
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  + (−0.498 + 2.18i)2-s + (−2.71 − 1.30i)4-s + 4.23·5-s + (−1.05 − 4.60i)7-s + (1.42 − 1.78i)8-s + (−2.10 + 9.24i)10-s + (−1.60 − 2.01i)11-s + (2.13 − 2.67i)13-s + 10.5·14-s + (−0.580 − 0.727i)16-s − 2.41·17-s + (2.03 − 0.982i)19-s + (−11.5 − 5.53i)20-s + (5.18 − 2.49i)22-s + (−1.31 − 5.75i)23-s + ⋯
L(s)  = 1  + (−0.352 + 1.54i)2-s + (−1.35 − 0.654i)4-s + 1.89·5-s + (−0.397 − 1.74i)7-s + (0.502 − 0.629i)8-s + (−0.666 + 2.92i)10-s + (−0.483 − 0.606i)11-s + (0.592 − 0.742i)13-s + 2.82·14-s + (−0.145 − 0.181i)16-s − 0.586·17-s + (0.467 − 0.225i)19-s + (−2.57 − 1.23i)20-s + (1.10 − 0.532i)22-s + (−0.274 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38293 + 0.449684i\)
\(L(\frac12)\) \(\approx\) \(1.38293 + 0.449684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.05 - 6.73i)T \)
good2 \( 1 + (0.498 - 2.18i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
7 \( 1 + (1.05 + 4.60i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (1.60 + 2.01i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.13 + 2.67i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 2.41T + 17T^{2} \)
19 \( 1 + (-2.03 + 0.982i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.31 + 5.75i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-3.93 - 1.89i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-1.17 + 1.47i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (0.455 - 1.99i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (5.66 - 7.10i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (0.855 - 3.74i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (4.01 - 1.93i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.0234 + 0.0112i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-5.13 - 6.44i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-0.316 + 1.38i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-2.71 - 1.30i)T + (41.7 + 52.3i)T^{2} \)
73 \( 1 + (0.404 - 1.77i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (2.64 - 3.31i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-1.30 - 1.64i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (11.3 - 5.48i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-8.92 - 11.1i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.21911428856432501224723064073, −9.822539039042526798874794195722, −8.724719265609595343439091995032, −7.990489037756862792455749044234, −6.76585235488262345051398438037, −6.46759404299344824138644996004, −5.54335319158725368429350799601, −4.62270140787761941236369292149, −2.90644677234028207731441664266, −0.924635404898606316901610450077, 1.82202097190678592957448027090, 2.19793271259017043761108435030, 3.26713557699627681178985572625, 4.98234462428121939426894088986, 5.82728352122824752358687550702, 6.66449618853990367183898689916, 8.603480601506807690845393302700, 9.110169103509509132662047917966, 9.758787644063066074744014047664, 10.25330914927866565101844318599

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