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Properties

Label 2-738-369.40-c1-0-36
Degree $2$
Conductor $738$
Sign $-0.190 + 0.981i$
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 + (1.57 − 0.727i)3-s + (−0.499 + 0.866i)4-s + (0.714 − 1.23i)5-s + (−1.41 − 0.997i)6-s + (2.06 − 1.19i)7-s + 0.999·8-s + (1.94 − 2.28i)9-s − 1.42·10-s + (3.59 − 2.07i)11-s + (−0.156 + 1.72i)12-s + (−3.86 − 2.23i)13-s + (−2.06 − 1.19i)14-s + (0.223 − 2.46i)15-s + (−0.5 − 0.866i)16-s + 8.04i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.907 − 0.419i)3-s + (−0.249 + 0.433i)4-s + (0.319 − 0.553i)5-s + (−0.577 − 0.407i)6-s + (0.779 − 0.450i)7-s + 0.353·8-s + (0.647 − 0.761i)9-s − 0.451·10-s + (1.08 − 0.625i)11-s + (−0.0451 + 0.497i)12-s + (−1.07 − 0.618i)13-s + (−0.551 − 0.318i)14-s + (0.0576 − 0.636i)15-s + (−0.125 − 0.216i)16-s + 1.95i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24902 - 1.51548i\)
\(L(\frac12)\) \(\approx\) \(1.24902 - 1.51548i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 0.727i)T \)
41 \( 1 + (-4.17 + 4.85i)T \)
good5 \( 1 + (-0.714 + 1.23i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.06 + 1.19i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.59 + 2.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.86 + 2.23i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 8.04iT - 17T^{2} \)
19 \( 1 - 0.0146iT - 19T^{2} \)
23 \( 1 + (1.70 - 2.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.69 - 2.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.83 + 6.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.09T + 37T^{2} \)
43 \( 1 + (4.92 + 8.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.3 - 6.54i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.16iT - 53T^{2} \)
59 \( 1 + (-6.49 + 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.312 + 0.541i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.40 - 3.69i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + 2.91T + 73T^{2} \)
79 \( 1 + (10.9 - 6.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.00 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.614iT - 89T^{2} \)
97 \( 1 + (-3.71 + 2.14i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.932478377105496712140524204075, −9.316823870501702770267842196440, −8.350047935758696123620226045339, −7.967759913968801920303767336570, −6.88285416646211338851548354826, −5.62624430912413316135415845917, −4.26647946907965637238415261650, −3.49666924411096740176583682474, −2.02648764533667797426560885087, −1.17765560737167739210918855959, 1.85739481931309652031659174791, 2.86557070631180910394114627543, 4.51377538703699813212622316962, 4.96938249543182284928608921788, 6.56426229633726818002093877544, 7.17130887014351943886336130712, 8.069295087739589671188559498526, 8.984536626172014483033229703528, 9.626573866183837581614391493787, 10.15058419099653445155601111176

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