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Properties

Label 2-740-740.447-c1-0-68
Degree $2$
Conductor $740$
Sign $0.542 + 0.839i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
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  + (−1.40 + 0.123i)2-s + (1.96 − 0.347i)4-s + (2.22 + 0.255i)5-s + (−2.73 + 0.732i)8-s + (−2.95 − 0.520i)9-s + (−3.16 − 0.0863i)10-s + (3.91 − 5.59i)13-s + (3.75 − 1.36i)16-s + (−3.67 − 5.24i)17-s + (4.22 + 0.369i)18-s + (4.46 − 0.267i)20-s + (4.86 + 1.13i)25-s + (−4.83 + 8.36i)26-s + (2.83 − 4.91i)29-s + (−5.12 + 2.39i)32-s + ⋯
L(s)  = 1  + (−0.996 + 0.0871i)2-s + (0.984 − 0.173i)4-s + (0.993 + 0.114i)5-s + (−0.965 + 0.258i)8-s + (−0.984 − 0.173i)9-s + (−0.999 − 0.0273i)10-s + (1.08 − 1.55i)13-s + (0.939 − 0.342i)16-s + (−0.891 − 1.27i)17-s + (0.996 + 0.0871i)18-s + (0.998 − 0.0599i)20-s + (0.973 + 0.227i)25-s + (−0.947 + 1.64i)26-s + (0.526 − 0.912i)29-s + (−0.906 + 0.422i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.542 + 0.839i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ 0.542 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898973 - 0.489345i\)
\(L(\frac12)\) \(\approx\) \(0.898973 - 0.489345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.123i)T \)
5 \( 1 + (-2.22 - 0.255i)T \)
37 \( 1 + (2.02 + 5.73i)T \)
good3 \( 1 + (2.95 + 0.520i)T^{2} \)
7 \( 1 + (4.49 - 5.36i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.91 + 5.59i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (3.67 + 5.24i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.83 + 4.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
41 \( 1 + (-1.89 - 10.7i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.16 + 6.78i)T + (-34.0 - 40.6i)T^{2} \)
59 \( 1 + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-13.3 + 2.35i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (43.0 - 51.3i)T^{2} \)
71 \( 1 + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (-12.0 + 12.0i)T - 73iT^{2} \)
79 \( 1 + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (28.3 - 77.9i)T^{2} \)
89 \( 1 + (3.68 - 1.34i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (9.81 + 2.62i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.09861740011519773445624994849, −9.374561296773073245452660951435, −8.598964771531232339228113071797, −7.895472935369146356564169510097, −6.66149298175383486922987135692, −5.99920511788941673427931440490, −5.20706352082559019419341704427, −3.18909300765167027775828859007, −2.37619155146204510498792685358, −0.74118783755542262232078071357, 1.50950456941985640873031232189, 2.43142114784722287994592611349, 3.84879736135216835443741193413, 5.43422159052525844688139747963, 6.36517143490838117594686860745, 6.84716312828115453216841651972, 8.528286121265992340005502648131, 8.626869225049548514874590138626, 9.514043854306053368832776841087, 10.54358878897592680178683166232

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