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Properties

Label 2-738-369.286-c1-0-23
Degree $2$
Conductor $738$
Sign $-0.304 + 0.952i$
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 + (−1.35 − 1.08i)3-s + (−0.499 − 0.866i)4-s + (−1.58 − 2.74i)5-s + (1.61 − 0.626i)6-s + (1.86 + 1.07i)7-s + 0.999·8-s + (0.645 + 2.92i)9-s + 3.17·10-s + (−0.206 − 0.119i)11-s + (−0.264 + 1.71i)12-s + (6.02 − 3.47i)13-s + (−1.86 + 1.07i)14-s + (−0.838 + 5.42i)15-s + (−0.5 + 0.866i)16-s − 4.96i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.779 − 0.626i)3-s + (−0.249 − 0.433i)4-s + (−0.709 − 1.22i)5-s + (0.659 − 0.255i)6-s + (0.704 + 0.406i)7-s + 0.353·8-s + (0.215 + 0.976i)9-s + 1.00·10-s + (−0.0623 − 0.0360i)11-s + (−0.0763 + 0.494i)12-s + (1.67 − 0.964i)13-s + (−0.497 + 0.287i)14-s + (−0.216 + 1.40i)15-s + (−0.125 + 0.216i)16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.304 + 0.952i$
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: \((2,\ 738,\ (\ :1/2),\ -0.304 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421550 - 0.577175i\)
\(L(\frac12)\) \(\approx\) \(0.421550 - 0.577175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.08i)T \)
41 \( 1 + (4.10 + 4.91i)T \)
good5 \( 1 + (1.58 + 2.74i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.86 - 1.07i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.206 + 0.119i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.02 + 3.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.96iT - 17T^{2} \)
19 \( 1 - 6.10iT - 19T^{2} \)
23 \( 1 + (1.71 + 2.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.10 - 1.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.95 + 8.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
43 \( 1 + (1.63 - 2.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.37 + 3.10i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.18iT - 53T^{2} \)
59 \( 1 + (-5.69 - 9.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.854 + 1.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.46 + 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.31iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + (-0.375 - 0.216i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.97 - 3.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.54iT - 89T^{2} \)
97 \( 1 + (-6.84 - 3.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.17665400734432016509235371001, −8.838612572433234886395446662604, −8.215870489733451199918843334161, −7.80366219588486292002068019495, −6.58315286032469532074120317360, −5.51665233462509990824397792621, −5.13369912260442462354682684762, −3.87918578145866826209006618580, −1.64779328139162205399674071724, −0.50779688207227551526191874552, 1.52358340394580771895169697094, 3.40434631885767960010577309784, 3.91228985778009887947139309040, 5.01092968686593346579737126883, 6.46358868649650532376506264685, 6.98637860741799803419069917902, 8.243555024443254882196904189960, 8.979344105317582148734262792213, 10.20658148739096418499117010388, 10.79378603111761337718362385129

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