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Properties

Label 2-1638-91.25-c1-0-20
Degree $2$
Conductor $1638$
Sign $0.332 + 0.943i$
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 + (−2.25 + 1.30i)5-s + (−1.49 + 2.18i)7-s + 0.999i·8-s + (1.30 − 2.25i)10-s + (−3.11 − 1.80i)11-s + (3.60 + 0.167i)13-s + (0.199 − 2.63i)14-s + (−0.5 − 0.866i)16-s + (−1.41 + 2.45i)17-s + (−1.15 + 0.667i)19-s + 2.60i·20-s + 3.60·22-s + (1.46 + 2.54i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.00 + 0.581i)5-s + (−0.563 + 0.825i)7-s + 0.353i·8-s + (0.411 − 0.712i)10-s + (−0.940 − 0.542i)11-s + (0.998 + 0.0463i)13-s + (0.0532 − 0.705i)14-s + (−0.125 − 0.216i)16-s + (−0.343 + 0.595i)17-s + (−0.265 + 0.153i)19-s + 0.581i·20-s + 0.767·22-s + (0.306 + 0.530i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.332 + 0.943i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.332 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2569481689\)
\(L(\frac12)\) \(\approx\) \(0.2569481689\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 - 2.18i)T \)
13 \( 1 + (-3.60 - 0.167i)T \)
good5 \( 1 + (2.25 - 1.30i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.11 + 1.80i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.41 - 2.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.15 - 0.667i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.46 - 2.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.97T + 29T^{2} \)
31 \( 1 + (3.46 + 2i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.144 - 0.0835i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.03iT - 41T^{2} \)
43 \( 1 - 2.33T + 43T^{2} \)
47 \( 1 + (-8.43 + 4.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.0 - 5.78i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.01 + 3.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 5.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.76iT - 71T^{2} \)
73 \( 1 + (9.93 + 5.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.37 + 5.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.10iT - 83T^{2} \)
89 \( 1 + (-7.07 + 4.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.139iT - 97T^{2} \)
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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

−8.951713620225498864888724858211, −8.514693776039975450326191834178, −7.63879927449396571433228452530, −7.03068967133564280931138429152, −5.94718075399196829867784207344, −5.52934563241527913935989449695, −3.95093708185205065199882821064, −3.23076453172593429965477085329, −2.04341558849532117051707766742, −0.15154815600555986353601338623, 0.949066713908425135246466670947, 2.48477370107954682694832265153, 3.67541481626905637568293811747, 4.25296270608237626132226493318, 5.34824807480832354131014279796, 6.61586637839134071658635463068, 7.40276426987736887384940601174, 7.933882091438570182701182755454, 8.793873833166353152268179555036, 9.450132895739185176984196766540

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