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Properties

Label 2-1638-91.81-c1-0-30
Degree $2$
Conductor $1638$
Sign $-0.342 + 0.939i$
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 + (0.441 + 0.764i)5-s + (−2.45 + 0.989i)7-s − 0.999·8-s + 0.882·10-s − 1.55·11-s + (2.13 + 2.90i)13-s + (−0.369 + 2.61i)14-s + (−0.5 + 0.866i)16-s + (−3.58 − 6.21i)17-s + 4.74·19-s + (0.441 − 0.764i)20-s + (−0.775 + 1.34i)22-s + (2.64 − 4.58i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.197 + 0.341i)5-s + (−0.927 + 0.374i)7-s − 0.353·8-s + 0.279·10-s − 0.467·11-s + (0.591 + 0.805i)13-s + (−0.0988 + 0.700i)14-s + (−0.125 + 0.216i)16-s + (−0.870 − 1.50i)17-s + 1.08·19-s + (0.0986 − 0.170i)20-s + (−0.165 + 0.286i)22-s + (0.551 − 0.955i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.342 + 0.939i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.490361308\)
\(L(\frac12)\) \(\approx\) \(1.490361308\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 - 0.989i)T \)
13 \( 1 + (-2.13 - 2.90i)T \)
good5 \( 1 + (-0.441 - 0.764i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
17 \( 1 + (3.58 + 6.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 + (-2.64 + 4.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.87 + 6.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.165 - 0.286i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.02 - 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.976 + 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.74 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.63 - 4.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 7.50T + 67T^{2} \)
71 \( 1 + (5.00 - 8.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.93 - 3.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (-7.40 + 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.79 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.340216662386752445675650788072, −8.633488033628491806753447524327, −7.38061272870319965519722054428, −6.58260295370337126487012712429, −5.90418346088316703767868937881, −4.88961740229015126542961979960, −4.00434977705801878607123188343, −2.84562631351381397659725318558, −2.35351543431817298437874159904, −0.54478454441223429797192167955, 1.28333925802083488876178153378, 3.05682294979603718537748281338, 3.64287802969711164642861830488, 4.82020618614155637696320159476, 5.65753866756505933865374583648, 6.27747535898707683965653681736, 7.25190073480391315912188767365, 7.83662123907271331630554007939, 8.970201296537730629837220673246, 9.267911339300811241672650147377

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