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Properties

Label 2-1638-91.81-c1-0-45
Degree $2$
Conductor $1638$
Sign $-0.411 - 0.911i$
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 + (−1.14 − 1.98i)5-s + (−1.12 − 2.39i)7-s − 0.999·8-s − 2.29·10-s − 0.878·11-s + (−0.786 + 3.51i)13-s + (−2.63 − 0.222i)14-s + (−0.5 + 0.866i)16-s + (−3.20 − 5.54i)17-s + 1.50·19-s + (−1.14 + 1.98i)20-s + (−0.439 + 0.760i)22-s + (0.658 − 1.14i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.512 − 0.887i)5-s + (−0.425 − 0.904i)7-s − 0.353·8-s − 0.724·10-s − 0.264·11-s + (−0.218 + 0.975i)13-s + (−0.704 − 0.0593i)14-s + (−0.125 + 0.216i)16-s + (−0.777 − 1.34i)17-s + 0.346·19-s + (−0.256 + 0.443i)20-s + (−0.0936 + 0.162i)22-s + (0.137 − 0.237i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.411 - 0.911i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4615619553\)
\(L(\frac12)\) \(\approx\) \(0.4615619553\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 + 2.39i)T \)
13 \( 1 + (0.786 - 3.51i)T \)
good5 \( 1 + (1.14 + 1.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.878T + 11T^{2} \)
17 \( 1 + (3.20 + 5.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.50T + 19T^{2} \)
23 \( 1 + (-0.658 + 1.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.669 - 1.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.94 - 3.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.69 - 8.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.80 + 3.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.95 - 8.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.188 - 0.327i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.22 + 2.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.98 + 5.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.81T + 61T^{2} \)
67 \( 1 - 9.75T + 67T^{2} \)
71 \( 1 + (1.02 - 1.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.432 + 0.749i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.18 + 7.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 + (6.41 - 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.40 + 7.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.988488950233110949636777982564, −8.183474679541196434050265585362, −7.09750930772882506200670119529, −6.56349280551938919130363886654, −5.05369955845632539053401495280, −4.68806429357893654540924267303, −3.78620171509677792321050629097, −2.78453657187582889643177857828, −1.38036062653787145110953187871, −0.15788906209981625401113486649, 2.26429670053348643629428087196, 3.23011744309070769780625940376, 3.95116089530870373609332517127, 5.26805091793364228200991556534, 5.85625185617381572993521548614, 6.73072174897674761746194992201, 7.41804777940943189090305889262, 8.255205598533723437227054737184, 8.891806815571275710195397737303, 9.922279398995192094098794916713

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