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Properties

Label 2-1638-13.10-c1-0-12
Degree $2$
Conductor $1638$
Sign $-0.473 - 0.880i$
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 + 3.48i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−1.74 − 3.02i)10-s + (2.32 − 1.34i)11-s + (−3.15 + 1.74i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (2.95 − 5.12i)17-s + (4.50 + 2.59i)19-s + (3.02 + 1.74i)20-s + (−1.34 + 2.32i)22-s + (3.52 + 6.10i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 1.55i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.551 − 0.955i)10-s + (0.700 − 0.404i)11-s + (−0.874 + 0.484i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.717 − 1.24i)17-s + (1.03 + 0.596i)19-s + (0.675 + 0.389i)20-s + (−0.285 + 0.495i)22-s + (0.735 + 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300901409\)
\(L(\frac12)\) \(\approx\) \(1.300901409\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.15 - 1.74i)T \)
good5 \( 1 - 3.48iT - 5T^{2} \)
11 \( 1 + (-2.32 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.50 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.52 - 6.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.56 - 6.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.782iT - 31T^{2} \)
37 \( 1 + (-6.76 + 3.90i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 - 0.0788i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.165 - 0.285i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.60iT - 47T^{2} \)
53 \( 1 + 3.92T + 53T^{2} \)
59 \( 1 + (8.26 + 4.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.70 - 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.837 + 0.483i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.62 + 2.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.0iT - 73T^{2} \)
79 \( 1 + 0.293T + 79T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 + (4.51 - 2.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.73 + 1.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.525102742815466528116759368825, −9.084583718519104710342126303960, −7.63558936013382788797138075922, −7.44589544266410787433849460596, −6.63867582166535453157116187292, −5.78684499884123563227566893026, −4.87856568786880463447939136909, −3.39127925123866402381072766280, −2.74883803919799337556115608951, −1.35678539728457361497009440933, 0.69917211380920880250532149126, 1.54754253100856410646313792759, 2.87202064354737723262237851712, 4.25075064277953914167896289138, 4.77643461800186061988156860404, 5.79132524959693989833754593616, 6.88432051802404109211866170578, 7.956841250660286134280746472270, 8.259814759773995758183005970564, 9.246428660932733963551999814314

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