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Properties

Label 2-738-369.40-c1-0-26
Degree $2$
Conductor $738$
Sign $0.893 + 0.448i$
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 + (−0.359 + 1.69i)3-s + (−0.499 + 0.866i)4-s + (−0.0761 + 0.131i)5-s + (1.64 − 0.535i)6-s + (1.94 − 1.12i)7-s + 0.999·8-s + (−2.74 − 1.21i)9-s + 0.152·10-s + (3.89 − 2.24i)11-s + (−1.28 − 1.15i)12-s + (−1.91 − 1.10i)13-s + (−1.94 − 1.12i)14-s + (−0.196 − 0.176i)15-s + (−0.5 − 0.866i)16-s − 3.10i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.207 + 0.978i)3-s + (−0.249 + 0.433i)4-s + (−0.0340 + 0.0590i)5-s + (0.672 − 0.218i)6-s + (0.735 − 0.424i)7-s + 0.353·8-s + (−0.913 − 0.406i)9-s + 0.0481·10-s + (1.17 − 0.677i)11-s + (−0.371 − 0.334i)12-s + (−0.532 − 0.307i)13-s + (−0.520 − 0.300i)14-s + (−0.0506 − 0.0455i)15-s + (−0.125 − 0.216i)16-s − 0.754i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $0.893 + 0.448i$
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: \((2,\ 738,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20364 - 0.285073i\)
\(L(\frac12)\) \(\approx\) \(1.20364 - 0.285073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.359 - 1.69i)T \)
41 \( 1 + (-6.11 + 1.90i)T \)
good5 \( 1 + (0.0761 - 0.131i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.94 + 1.12i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.89 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.91 + 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 + 0.0605iT - 19T^{2} \)
23 \( 1 + (-0.0253 + 0.0438i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.83 + 2.79i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.00164 + 0.00284i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.348T + 37T^{2} \)
43 \( 1 + (-1.74 - 3.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.96 + 5.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.876iT - 53T^{2} \)
59 \( 1 + (-2.14 + 3.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.86 - 4.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.29 - 0.748i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 - 5.01T + 73T^{2} \)
79 \( 1 + (-0.914 + 0.527i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.14 + 5.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.00331iT - 89T^{2} \)
97 \( 1 + (3.02 - 1.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39549949271289874640780715752, −9.493861510046029255941370228493, −8.905594642825115702250172220496, −7.980720114943665485687948651383, −6.89088870455752385446553315458, −5.62118632906615825507240353245, −4.62566048631042259093856635215, −3.82371644167232672165938760360, −2.73223719351691843978203813534, −0.916799101437371605677803808732, 1.23169612149304830045311633424, 2.33756973359601276816880219376, 4.26259274068871739671623949883, 5.24981611073313707996170798723, 6.30002973223097346956094429464, 6.91365508534049850707444367835, 7.81699754624854082407692419870, 8.569898767334739355645896328707, 9.281908400451240576621395670810, 10.43526463213842637035578755009

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