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Properties

Label 2-738-369.286-c1-0-25
Degree $2$
Conductor $738$
Sign $0.520 + 0.854i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $0.520 + 0.854i$
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: \((2,\ 738,\ (\ :1/2),\ 0.520 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421244 - 0.236714i\)
\(L(\frac12)\) \(\approx\) \(0.421244 - 0.236714i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 6.24i)T \)
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

−10.11380505339595754545156730693, −9.437883882147408884172655676475, −8.474141716935484635248323988991, −7.889469528945083026569944904098, −6.70581923642178437770287266719, −5.71313930055189599646037791700, −4.94116583588312294745765163308, −3.74794278643837478435043183625, −2.75519222268486790767393867622, −0.26348651900245857499316399944, 1.61693175671266579375656826706, 2.67545272777955307457151279143, 3.62371482943248143446741939972, 5.20817028669813908558938741983, 6.25057926645255558504509516213, 7.19825098405377300860363036474, 7.979363805582723155002869412417, 8.847780314436563555951380050839, 9.572074304759189004823178807388, 10.59818297648369612696958973429

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