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Properties

Label 2-1638-91.81-c1-0-5
Degree $2$
Conductor $1638$
Sign $0.0131 - 0.999i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.924 + 1.60i)5-s + (−1.65 − 2.06i)7-s − 0.999·8-s + 1.84·10-s − 0.715·11-s + (−2.81 + 2.25i)13-s + (−2.61 + 0.405i)14-s + (−0.5 + 0.866i)16-s + (2.15 + 3.73i)17-s − 6.87·19-s + (0.924 − 1.60i)20-s + (−0.357 + 0.619i)22-s + (−3.58 + 6.20i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.413 + 0.716i)5-s + (−0.626 − 0.779i)7-s − 0.353·8-s + 0.584·10-s − 0.215·11-s + (−0.780 + 0.625i)13-s + (−0.698 + 0.108i)14-s + (−0.125 + 0.216i)16-s + (0.522 + 0.905i)17-s − 1.57·19-s + (0.206 − 0.358i)20-s + (−0.0762 + 0.132i)22-s + (−0.746 + 1.29i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0131 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0131 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.0131 - 0.999i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.0131 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8278199948\)
\(L(\frac12)\) \(\approx\) \(0.8278199948\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (1.65 + 2.06i)T \)
13 \( 1 + (2.81 - 2.25i)T \)
good5 \( 1 + (-0.924 - 1.60i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.715T + 11T^{2} \)
17 \( 1 + (-2.15 - 3.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 6.87T + 19T^{2} \)
23 \( 1 + (3.58 - 6.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.63 - 8.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.11 + 1.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.06 - 1.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.23 + 3.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0979 - 0.169i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.60 + 7.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.80 - 6.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.24 + 2.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.62T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + (7.50 - 13.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.989 - 1.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.37 - 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.71T + 83T^{2} \)
89 \( 1 + (2.23 - 3.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.39 + 2.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.967370116760660630855764377012, −8.977905386042173994754446837907, −7.991741558270680993158739768918, −6.90399558972667385000351006101, −6.45769999670523407539929964438, −5.46776938923115297795072719528, −4.35844756299900023346991653408, −3.60805897749544897409294567373, −2.64004746913072041423641712347, −1.60771683180629157993003839379, 0.26238334180716385033901998755, 2.24646489506679352127267186895, 3.09154317828813335927157636990, 4.51928854396651200124840132695, 5.03557955921317524836770695147, 6.04350937725774146926741456901, 6.47026297621202774174746906544, 7.67021183006997459724097879184, 8.367418999826527900233932878016, 9.062882422789693865771160451311

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