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Properties

Label 2-738-9.4-c1-0-5
Degree $2$
Conductor $738$
Sign $-0.603 - 0.797i$
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 + (1.19 + 1.25i)3-s + (−0.499 + 0.866i)4-s + (−1.81 + 3.14i)5-s + (0.490 − 1.66i)6-s + (−0.330 − 0.571i)7-s + 0.999·8-s + (−0.151 + 2.99i)9-s + 3.62·10-s + (2.36 + 4.10i)11-s + (−1.68 + 0.405i)12-s + (1.96 − 3.41i)13-s + (−0.330 + 0.571i)14-s + (−6.10 + 1.47i)15-s + (−0.5 − 0.866i)16-s − 5.36·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.689 + 0.724i)3-s + (−0.249 + 0.433i)4-s + (−0.811 + 1.40i)5-s + (0.200 − 0.678i)6-s + (−0.124 − 0.216i)7-s + 0.353·8-s + (−0.0503 + 0.998i)9-s + 1.14·10-s + (0.714 + 1.23i)11-s + (−0.486 + 0.117i)12-s + (0.546 − 0.946i)13-s + (−0.0882 + 0.152i)14-s + (−1.57 + 0.380i)15-s + (−0.125 − 0.216i)16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.603 - 0.797i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.603 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443047 + 0.890786i\)
\(L(\frac12)\) \(\approx\) \(0.443047 + 0.890786i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.19 - 1.25i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.81 - 3.14i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.330 + 0.571i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 4.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.96 + 3.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.36T + 17T^{2} \)
19 \( 1 + 5.47T + 19T^{2} \)
23 \( 1 + (-3.52 + 6.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.45 - 7.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.83 - 8.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.71T + 37T^{2} \)
43 \( 1 + (1.03 + 1.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.32 + 4.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.71T + 53T^{2} \)
59 \( 1 + (3.23 - 5.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.72 + 4.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.42T + 71T^{2} \)
73 \( 1 + 2.58T + 73T^{2} \)
79 \( 1 + (-4.88 - 8.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.23 - 7.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.36T + 89T^{2} \)
97 \( 1 + (5.34 + 9.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.57299534353797820917619941290, −10.22334419165248068083521374385, −8.794715559795023703595357361854, −8.523860187925727710186896726372, −7.08330735499457701320792823895, −6.83283162966707894356414835551, −4.80121745779145320058412569238, −3.92309064004683481372914378389, −3.17041148678778910226465183816, −2.16527462280590685484369754307, 0.52619265142247903577084219011, 1.83809963645597188696742475679, 3.72120487164473337308924939722, 4.46065831745367299120624465460, 5.92511759276946263389163423690, 6.60785392192296983514304681189, 7.69610489785448818020471794127, 8.538457837392866985099345327028, 8.861338678908151177987656859076, 9.424265939810202846460454641960

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