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Properties

Label 2-738-41.40-c1-0-1
Degree $2$
Conductor $738$
Sign $-0.199 - 0.979i$
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  + 2-s + 4-s − 4.27·5-s − 2i·7-s + 8-s − 4.27·10-s + 4i·11-s + 4.27i·13-s − 2i·14-s + 16-s + 6.27i·17-s + 6.27i·19-s − 4.27·20-s + 4i·22-s + 13.2·25-s + 4.27i·26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.91·5-s − 0.755i·7-s + 0.353·8-s − 1.35·10-s + 1.20i·11-s + 1.18i·13-s − 0.534i·14-s + 0.250·16-s + 1.52i·17-s + 1.43i·19-s − 0.955·20-s + 0.852i·22-s + 2.65·25-s + 0.838i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.199 - 0.979i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.199 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729603 + 0.892748i\)
\(L(\frac12)\) \(\approx\) \(0.729603 + 0.892748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
41 \( 1 + (1.27 + 6.27i)T \)
good5 \( 1 + 4.27T + 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 4.27iT - 13T^{2} \)
17 \( 1 - 6.27iT - 17T^{2} \)
19 \( 1 - 6.27iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6.54iT - 29T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 - 2.54iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 2.27T + 59T^{2} \)
61 \( 1 + 6.54T + 61T^{2} \)
67 \( 1 - 1.72iT - 67T^{2} \)
71 \( 1 - 3.72iT - 71T^{2} \)
73 \( 1 - 0.274T + 73T^{2} \)
79 \( 1 + 6.54iT - 79T^{2} \)
83 \( 1 - 6.27T + 83T^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 - 0.549iT - 97T^{2} \)
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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

−10.73325489841058080359300442581, −10.07713406056604444116735468301, −8.631828901267069459741375521646, −7.79178637437816718491693511866, −7.20730830704759436184571451709, −6.36467108543336768451781308691, −4.82100130013478974467469849290, −3.96136705892344774785044281909, −3.72848067465408593127622399440, −1.77304287294618449852133934945, 0.47185678393638216328726749294, 3.03921360710154388789808561906, 3.29849242470671372853185714123, 4.77161400987192467133138022766, 5.32773746847089667115698774035, 6.72191412550350711063132886544, 7.45862718969343034702582694988, 8.375576232923724443608983213584, 8.954516554343402465561531920916, 10.51035014208524928805407186059

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