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Properties

Label 2-1638-91.51-c1-0-38
Degree $2$
Conductor $1638$
Sign $0.871 - 0.490i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (2.18 + 1.26i)5-s + (1.47 − 2.19i)7-s + 0.999i·8-s + (1.26 + 2.18i)10-s + (4.88 − 2.82i)11-s + (−3.13 + 1.78i)13-s + (2.37 − 1.16i)14-s + (−0.5 + 0.866i)16-s + (−0.123 − 0.214i)17-s + (2.70 + 1.56i)19-s + 2.52i·20-s + 5.64·22-s + (1.80 − 3.11i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.976 + 0.563i)5-s + (0.556 − 0.830i)7-s + 0.353i·8-s + (0.398 + 0.690i)10-s + (1.47 − 0.850i)11-s + (−0.869 + 0.493i)13-s + (0.634 − 0.311i)14-s + (−0.125 + 0.216i)16-s + (−0.0300 − 0.0519i)17-s + (0.620 + 0.358i)19-s + 0.563i·20-s + 1.20·22-s + (0.375 − 0.650i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.323509946\)
\(L(\frac12)\) \(\approx\) \(3.323509946\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.47 + 2.19i)T \)
13 \( 1 + (3.13 - 1.78i)T \)
good5 \( 1 + (-2.18 - 1.26i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.88 + 2.82i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.123 + 0.214i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.70 - 1.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.80 + 3.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + (5.30 - 3.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.01 + 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 - 5.56T + 43T^{2} \)
47 \( 1 + (-5.78 - 3.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.28 - 5.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.89 - 3.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.39 - 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.09i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.06iT - 71T^{2} \)
73 \( 1 + (-6.81 + 3.93i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.22 + 5.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.662iT - 83T^{2} \)
89 \( 1 + (4.86 + 2.81i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.8iT - 97T^{2} \)
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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

−9.280928623256186749772328939554, −8.791349087730665969005798522658, −7.47812867779085870482094511461, −6.99558357667702835191271971991, −6.20740632673236425508058776775, −5.46449297286156389567502800793, −4.42169087441776567386573157913, −3.64739343458266774562040236971, −2.50322736386390718843831897675, −1.31519497827798729815440284205, 1.39136501508414463369010968506, 2.09049905965539102848108306459, 3.24035728148014023280102663890, 4.54600424443631369379860366169, 5.12043997832125658012239870747, 5.83169858022709445080565964475, 6.72695991454860571974915540265, 7.64044059835103243972598311104, 8.823359141434014416775482268616, 9.476927619813795088861513554903

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