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Properties

Label 2-1638-273.68-c1-0-21
Degree $2$
Conductor $1638$
Sign $0.467 + 0.884i$
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  − i·2-s − 4-s + (0.902 − 1.56i)5-s + (1.34 + 2.27i)7-s + i·8-s + (−1.56 − 0.902i)10-s + (−0.655 − 0.378i)11-s + (−3.49 + 0.872i)13-s + (2.27 − 1.34i)14-s + 16-s + 5.09·17-s + (1.28 − 0.742i)19-s + (−0.902 + 1.56i)20-s + (−0.378 + 0.655i)22-s − 0.139i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.403 − 0.699i)5-s + (0.507 + 0.861i)7-s + 0.353i·8-s + (−0.494 − 0.285i)10-s + (−0.197 − 0.114i)11-s + (−0.970 + 0.241i)13-s + (0.609 − 0.358i)14-s + 0.250·16-s + 1.23·17-s + (0.294 − 0.170i)19-s + (−0.201 + 0.349i)20-s + (−0.0806 + 0.139i)22-s − 0.0291i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.467 + 0.884i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.467 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844500391\)
\(L(\frac12)\) \(\approx\) \(1.844500391\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-1.34 - 2.27i)T \)
13 \( 1 + (3.49 - 0.872i)T \)
good5 \( 1 + (-0.902 + 1.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.655 + 0.378i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.09T + 17T^{2} \)
19 \( 1 + (-1.28 + 0.742i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.139iT - 23T^{2} \)
29 \( 1 + (-7.49 + 4.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.91 + 3.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 + (0.316 + 0.547i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.39 + 4.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.64 - 6.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.44 + 5.45i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.51T + 59T^{2} \)
61 \( 1 + (-8.72 + 5.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.03 - 8.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.93 - 5.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.14 + 5.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.71 - 2.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 + (12.3 + 7.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.500866112731150999922351153381, −8.456577659905423282842676296424, −8.043163134499779242991918829421, −6.83720845121609966950502416802, −5.55715102938662762793668227696, −5.19455215620824635537436319745, −4.28106917488241816936010189155, −2.95222950255015790625616676944, −2.13815907617343854927468278176, −0.946132260822605229057795451158, 1.04478342643031568066075558425, 2.61906709747484692719836221189, 3.62805541948719028100140464807, 4.81531950560330718573465454340, 5.35304569207444445222093558560, 6.54335838648672976792035062637, 7.05119213316496179200680970620, 7.83732935342536055957965207828, 8.451705703875790202322649278936, 9.671331332048895612938862137191

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