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Properties

Label 2-1638-91.9-c1-0-33
Degree $2$
Conductor $1638$
Sign $-0.295 + 0.955i$
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 + (0.769 − 1.33i)5-s + (−0.131 − 2.64i)7-s + 0.999·8-s − 1.53·10-s + 6.38·11-s + (0.520 − 3.56i)13-s + (−2.22 + 1.43i)14-s + (−0.5 − 0.866i)16-s + (2.19 − 3.79i)17-s + 0.101·19-s + (0.769 + 1.33i)20-s + (−3.19 − 5.52i)22-s + (4.54 + 7.87i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.344 − 0.596i)5-s + (−0.0498 − 0.998i)7-s + 0.353·8-s − 0.486·10-s + 1.92·11-s + (0.144 − 0.989i)13-s + (−0.593 + 0.383i)14-s + (−0.125 − 0.216i)16-s + (0.531 − 0.920i)17-s + 0.0233·19-s + (0.172 + 0.298i)20-s + (−0.680 − 1.17i)22-s + (0.947 + 1.64i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.295 + 0.955i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.701787185\)
\(L(\frac12)\) \(\approx\) \(1.701787185\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.131 + 2.64i)T \)
13 \( 1 + (-0.520 + 3.56i)T \)
good5 \( 1 + (-0.769 + 1.33i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 6.38T + 11T^{2} \)
17 \( 1 + (-2.19 + 3.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 0.101T + 19T^{2} \)
23 \( 1 + (-4.54 - 7.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.51 - 6.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.611 + 1.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.92 + 3.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.42 + 4.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.877 + 1.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.07 + 3.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.11 - 7.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.56 + 6.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 6.16T + 61T^{2} \)
67 \( 1 + 1.71T + 67T^{2} \)
71 \( 1 + (4.57 + 7.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.82 - 8.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.03 - 3.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.02T + 83T^{2} \)
89 \( 1 + (7.77 + 13.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.996 - 1.72i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.283595848082925844253313624547, −8.677782086978567413999233198766, −7.39601966068699949036608448129, −7.11279825503905831349700907825, −5.75446931183983492153194194406, −4.94270149262547587154181044014, −3.79397378406533865712860322692, −3.27783737255337668785587907836, −1.53176993554607895482884168071, −0.895720751784865718941226243714, 1.37579354158246629348839742167, 2.48880310974907243489745386421, 3.81348962402027047082729602299, 4.70325995608253562888738196432, 5.99702913836914790164901193023, 6.39286174967912565450330861030, 6.94871584110106032150243764650, 8.200543395531003740927749475962, 8.886334434642158864902810507415, 9.366449986310839335165766488507

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