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Properties

Label 2-738-123.101-c1-0-11
Degree $2$
Conductor $738$
Sign $-0.890 + 0.454i$
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  + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.589 − 3.71i)5-s + (3.08 − 2.63i)7-s + (0.987 + 0.156i)8-s + (−3.04 + 2.21i)10-s + (−1.49 − 0.915i)11-s + (4.02 − 0.316i)13-s + (−3.74 − 1.55i)14-s + (−0.309 − 0.951i)16-s + (−2.51 + 0.604i)17-s + (−0.625 − 0.0492i)19-s + (3.35 + 1.70i)20-s + (−0.137 + 1.74i)22-s + (1.23 − 3.81i)23-s + ⋯
L(s)  = 1  + (−0.321 − 0.630i)2-s + (−0.293 + 0.404i)4-s + (−0.263 − 1.66i)5-s + (1.16 − 0.995i)7-s + (0.349 + 0.0553i)8-s + (−0.963 + 0.700i)10-s + (−0.450 − 0.276i)11-s + (1.11 − 0.0878i)13-s + (−1.00 − 0.414i)14-s + (−0.0772 − 0.237i)16-s + (−0.610 + 0.146i)17-s + (−0.143 − 0.0112i)19-s + (0.750 + 0.382i)20-s + (−0.0293 + 0.372i)22-s + (0.258 − 0.794i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293113 - 1.22072i\)
\(L(\frac12)\) \(\approx\) \(0.293113 - 1.22072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 + 0.891i)T \)
3 \( 1 \)
41 \( 1 + (3.76 - 5.18i)T \)
good5 \( 1 + (0.589 + 3.71i)T + (-4.75 + 1.54i)T^{2} \)
7 \( 1 + (-3.08 + 2.63i)T + (1.09 - 6.91i)T^{2} \)
11 \( 1 + (1.49 + 0.915i)T + (4.99 + 9.80i)T^{2} \)
13 \( 1 + (-4.02 + 0.316i)T + (12.8 - 2.03i)T^{2} \)
17 \( 1 + (2.51 - 0.604i)T + (15.1 - 7.71i)T^{2} \)
19 \( 1 + (0.625 + 0.0492i)T + (18.7 + 2.97i)T^{2} \)
23 \( 1 + (-1.23 + 3.81i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-9.24 - 2.21i)T + (25.8 + 13.1i)T^{2} \)
31 \( 1 + (-4.07 - 5.60i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (8.27 + 6.01i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (6.60 - 3.36i)T + (25.2 - 34.7i)T^{2} \)
47 \( 1 + (4.17 - 4.88i)T + (-7.35 - 46.4i)T^{2} \)
53 \( 1 + (-2.57 + 10.7i)T + (-47.2 - 24.0i)T^{2} \)
59 \( 1 + (3.87 + 1.26i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.29 + 4.50i)T + (-35.8 - 49.3i)T^{2} \)
67 \( 1 + (3.42 - 2.09i)T + (30.4 - 59.6i)T^{2} \)
71 \( 1 + (0.573 - 0.935i)T + (-32.2 - 63.2i)T^{2} \)
73 \( 1 + (6.20 + 6.20i)T + 73iT^{2} \)
79 \( 1 + (-9.71 + 4.02i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + 5.82iT - 83T^{2} \)
89 \( 1 + (-4.41 - 5.16i)T + (-13.9 + 87.9i)T^{2} \)
97 \( 1 + (-3.55 - 5.80i)T + (-44.0 + 86.4i)T^{2} \)
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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

−10.23356855140774731820739614788, −8.849939612159040195614256556884, −8.456776155181256047113247871578, −7.922334864613335167899061119501, −6.55903464294749877868885925280, −4.91219028779031853281284651468, −4.68926756068618086765661101962, −3.51107536107720694937841604902, −1.64867382794436400898445631549, −0.77950336220301689949751124283, 1.95273940363793313156431297992, 3.10915369387108387454150059971, 4.50398430769725438207860100550, 5.61469104351964704649560390512, 6.47962008790588345108527438787, 7.22367071812432272865018805844, 8.216699797174325313682810963109, 8.681700918473829820124844181982, 10.01382291285165760955301781172, 10.67340185275762802856264154352

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