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Properties

Label 2-738-123.89-c1-0-6
Degree $2$
Conductor $738$
Sign $0.745 + 0.666i$
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  + (−0.891 + 0.453i)2-s + (0.587 − 0.809i)4-s + (2.73 − 0.433i)5-s + (−2.05 − 0.161i)7-s + (−0.156 + 0.987i)8-s + (−2.24 + 1.62i)10-s + (−1.44 − 6.00i)11-s + (3.47 + 2.96i)13-s + (1.90 − 0.788i)14-s + (−0.309 − 0.951i)16-s + (1.10 − 1.80i)17-s + (0.948 − 0.810i)19-s + (1.25 − 2.46i)20-s + (4.00 + 4.69i)22-s + (0.979 − 3.01i)23-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (0.293 − 0.404i)4-s + (1.22 − 0.193i)5-s + (−0.776 − 0.0610i)7-s + (−0.0553 + 0.349i)8-s + (−0.708 + 0.514i)10-s + (−0.434 − 1.80i)11-s + (0.963 + 0.822i)13-s + (0.508 − 0.210i)14-s + (−0.0772 − 0.237i)16-s + (0.267 − 0.436i)17-s + (0.217 − 0.185i)19-s + (0.281 − 0.551i)20-s + (0.854 + 1.00i)22-s + (0.204 − 0.628i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ 0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15308 - 0.440548i\)
\(L(\frac12)\) \(\approx\) \(1.15308 - 0.440548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 \)
41 \( 1 + (-3.19 - 5.54i)T \)
good5 \( 1 + (-2.73 + 0.433i)T + (4.75 - 1.54i)T^{2} \)
7 \( 1 + (2.05 + 0.161i)T + (6.91 + 1.09i)T^{2} \)
11 \( 1 + (1.44 + 6.00i)T + (-9.80 + 4.99i)T^{2} \)
13 \( 1 + (-3.47 - 2.96i)T + (2.03 + 12.8i)T^{2} \)
17 \( 1 + (-1.10 + 1.80i)T + (-7.71 - 15.1i)T^{2} \)
19 \( 1 + (-0.948 + 0.810i)T + (2.97 - 18.7i)T^{2} \)
23 \( 1 + (-0.979 + 3.01i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.66 + 5.97i)T + (-13.1 + 25.8i)T^{2} \)
31 \( 1 + (1.22 + 1.68i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-5.55 - 4.03i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (1.51 + 2.98i)T + (-25.2 + 34.7i)T^{2} \)
47 \( 1 + (0.225 + 2.86i)T + (-46.4 + 7.35i)T^{2} \)
53 \( 1 + (-8.43 + 5.16i)T + (24.0 - 47.2i)T^{2} \)
59 \( 1 + (-2.40 - 0.782i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.95 + 3.54i)T + (35.8 + 49.3i)T^{2} \)
67 \( 1 + (0.506 - 2.10i)T + (-59.6 - 30.4i)T^{2} \)
71 \( 1 + (-12.5 + 3.02i)T + (63.2 - 32.2i)T^{2} \)
73 \( 1 + (-3.65 + 3.65i)T - 73iT^{2} \)
79 \( 1 + (-13.1 - 5.44i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + 9.81iT - 83T^{2} \)
89 \( 1 + (0.782 - 9.94i)T + (-87.9 - 13.9i)T^{2} \)
97 \( 1 + (-18.0 - 4.32i)T + (86.4 + 44.0i)T^{2} \)
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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

−10.03523160567790215065549421515, −9.381236241774450885602339205279, −8.737776325011073630380963665605, −7.84767015594389863336177151492, −6.41436668810549633118055584461, −6.17094337427222880187769781947, −5.22848837752754848431781422348, −3.56701733886797794788319881522, −2.36171888949771875201263793626, −0.824025761000558594976391395173, 1.52200842570882355007810274380, 2.56926253667469770605458730253, 3.72611745284216919968938190375, 5.28667040805348816906962788466, 6.10813212210314650816126895416, 7.07043841244927570160865420880, 7.88372009867173126894119833769, 9.171153809272836206255044585105, 9.616372563043211214504468216969, 10.35364126055050343386955409292

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