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Properties

Label 2-1638-13.4-c1-0-13
Degree $2$
Conductor $1638$
Sign $0.311 - 0.950i$
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 − 0.332i·5-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.166 − 0.288i)10-s + (−2.26 − 1.30i)11-s + (3.41 + 1.16i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (1.94 + 3.36i)17-s + (4.85 − 2.80i)19-s + (0.288 − 0.166i)20-s + (−1.30 − 2.26i)22-s + (−2.10 + 3.64i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.148i·5-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (0.0526 − 0.0911i)10-s + (−0.681 − 0.393i)11-s + (0.946 + 0.323i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.471 + 0.816i)17-s + (1.11 − 0.643i)19-s + (0.0644 − 0.0372i)20-s + (−0.278 − 0.481i)22-s + (−0.439 + 0.760i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.351053428\)
\(L(\frac12)\) \(\approx\) \(2.351053428\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 - 1.16i)T \)
good5 \( 1 + 0.332iT - 5T^{2} \)
11 \( 1 + (2.26 + 1.30i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.94 - 3.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.85 + 2.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.10 - 3.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.593 - 1.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.07iT - 31T^{2} \)
37 \( 1 + (-0.499 - 0.288i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.451 + 0.260i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.53 - 2.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.0iT - 47T^{2} \)
53 \( 1 - 9.71T + 53T^{2} \)
59 \( 1 + (9.07 - 5.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.71 + 6.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 5.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.818 + 0.472i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.66iT - 73T^{2} \)
79 \( 1 + 0.943T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 + (2.92 + 1.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.03 - 2.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.377313244684780833146798927034, −8.668825064834070714053365438228, −7.88224074426394905918903894761, −7.07322499525216528596329389030, −6.14765154880555819688879936364, −5.53358998092128056145189061409, −4.66836609041800098213331958733, −3.54162698824933471898446459937, −2.89452496836120799673419166325, −1.33471838791788171415647508572, 0.819170151948762855511968623689, 2.30616552449167365414273615638, 3.23990902169170392037671910448, 4.04969479874292474326173540571, 5.15710476354905533571861542277, 5.77966528849822350646334928410, 6.74940014561860881513918390024, 7.54038928009900666139134357339, 8.367750895887518300530450217858, 9.467122341450245322681825679221

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