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Properties

Label 2-740-740.243-c1-0-15
Degree $2$
Conductor $740$
Sign $-0.419 - 0.907i$
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  + (−0.597 − 1.28i)2-s + (−1.28 + 1.53i)4-s + (−0.133 + 2.23i)5-s + (2.73 + 0.732i)8-s + (1.92 + 2.29i)9-s + (2.94 − 1.16i)10-s + (−5.39 + 0.472i)13-s + (−0.694 − 3.93i)16-s + (−6.20 − 0.542i)17-s + (1.79 − 3.84i)18-s + (−3.24 − 3.07i)20-s + (−4.96 − 0.598i)25-s + (3.83 + 6.63i)26-s + (−5.11 − 8.86i)29-s + (−4.63 + 3.24i)32-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)2-s + (−0.642 + 0.766i)4-s + (−0.0599 + 0.998i)5-s + (0.965 + 0.258i)8-s + (0.642 + 0.766i)9-s + (0.930 − 0.367i)10-s + (−1.49 + 0.130i)13-s + (−0.173 − 0.984i)16-s + (−1.50 − 0.131i)17-s + (0.422 − 0.906i)18-s + (−0.726 − 0.687i)20-s + (−0.992 − 0.119i)25-s + (0.751 + 1.30i)26-s + (−0.949 − 1.64i)29-s + (−0.819 + 0.573i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ -0.419 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241540 + 0.377573i\)
\(L(\frac12)\) \(\approx\) \(0.241540 + 0.377573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.597 + 1.28i)T \)
5 \( 1 + (0.133 - 2.23i)T \)
37 \( 1 + (3.09 - 5.23i)T \)
good3 \( 1 + (-1.92 - 2.29i)T^{2} \)
7 \( 1 + (2.39 - 6.57i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.39 - 0.472i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (6.20 + 0.542i)T + (16.7 + 2.95i)T^{2} \)
19 \( 1 + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.11 + 8.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
41 \( 1 + (-6.47 - 5.43i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.35 - 11.9i)T + (-18.1 - 49.8i)T^{2} \)
59 \( 1 + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-9.88 + 11.7i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (7.02 + 7.02i)T + 73iT^{2} \)
79 \( 1 + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (-3.03 - 17.2i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-17.9 + 4.82i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.71290635325925254896016817016, −9.834232647898970537415911469156, −9.334978307813448341302674320308, −7.928877399734802644046075805134, −7.46885928513647631598730468200, −6.47020530801877741320177524017, −4.86866115410691511887568140637, −4.13014422751497513690543997234, −2.71828932344090025872434802790, −2.04478249917241015769370133156, 0.25316688337843912529270627095, 1.84323438296325516963246059442, 3.96300049046089836315680734224, 4.81632547476745070231491920913, 5.60018992029321190826991939451, 6.85569874492245633418922848616, 7.34107224694336240897874333270, 8.503407664943808844536826975312, 9.152266233386380882399166018343, 9.728559906643936403969490864328

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