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Properties

Label 2-738-369.286-c1-0-9
Degree $2$
Conductor $738$
Sign $-0.841 - 0.540i$
Analytic cond. $5.89295$
Root an. cond. $2.42754$
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 + (0.528 + 1.64i)3-s + (−0.499 − 0.866i)4-s + (1.55 + 2.70i)5-s + (−1.69 − 0.366i)6-s + (0.0882 + 0.0509i)7-s + 0.999·8-s + (−2.44 + 1.74i)9-s − 3.11·10-s + (4.55 + 2.63i)11-s + (1.16 − 1.28i)12-s + (4.23 − 2.44i)13-s + (−0.0882 + 0.0509i)14-s + (−3.62 + 4.00i)15-s + (−0.5 + 0.866i)16-s + 5.23i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.305 + 0.952i)3-s + (−0.249 − 0.433i)4-s + (0.697 + 1.20i)5-s + (−0.691 − 0.149i)6-s + (0.0333 + 0.0192i)7-s + 0.353·8-s + (−0.813 + 0.581i)9-s − 0.986·10-s + (1.37 + 0.793i)11-s + (0.335 − 0.370i)12-s + (1.17 − 0.677i)13-s + (−0.0235 + 0.0136i)14-s + (−0.937 + 1.03i)15-s + (−0.125 + 0.216i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465610 + 1.58783i\)
\(L(\frac12)\) \(\approx\) \(0.465610 + 1.58783i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.528 - 1.64i)T \)
41 \( 1 + (0.195 + 6.40i)T \)
good5 \( 1 + (-1.55 - 2.70i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.0882 - 0.0509i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.55 - 2.63i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.23 + 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.23iT - 17T^{2} \)
19 \( 1 + 2.72iT - 19T^{2} \)
23 \( 1 + (2.45 + 4.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.03 - 0.599i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.11 - 3.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.60T + 37T^{2} \)
43 \( 1 + (-4.14 + 7.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.09 - 2.36i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.74iT - 53T^{2} \)
59 \( 1 + (3.11 + 5.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.38 - 12.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.33 + 1.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.40iT - 71T^{2} \)
73 \( 1 + 5.13T + 73T^{2} \)
79 \( 1 + (7.18 + 4.14i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.63 - 8.02i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.05iT - 89T^{2} \)
97 \( 1 + (1.48 + 0.854i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.42769981836541424381348877318, −10.02111545505352947796837837476, −8.902507022023301324710991424032, −8.459490731517305540336554961913, −7.07410741669836229859939593666, −6.37514984826964069590276524871, −5.60360907679127923119153768003, −4.25810801271378821898568839158, −3.37063166778766558816367176890, −1.93669776598178735003752296083, 1.07045361298736172612454424393, 1.65335740776798232801225951302, 3.20274303536663703014831156091, 4.32757302804385600570271901826, 5.75416281992783201962015756146, 6.41950623302664697525728246567, 7.66682911159248398686698871983, 8.602951078270096407955005787965, 9.116217564691248919402646554547, 9.634898194720738957767706292235

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