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Properties

Label 2-735-105.53-c1-0-56
Degree $2$
Conductor $735$
Sign $-0.204 + 0.978i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
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  + (0.455 − 1.69i)2-s + (1.67 + 0.421i)3-s + (−0.948 − 0.547i)4-s + (0.445 − 2.19i)5-s + (1.48 − 2.66i)6-s + (1.12 − 1.12i)8-s + (2.64 + 1.41i)9-s + (−3.52 − 1.75i)10-s + (−1.34 − 0.776i)11-s + (−1.36 − 1.32i)12-s + (4.50 + 4.50i)13-s + (1.67 − 3.49i)15-s + (−2.49 − 4.32i)16-s + (2.91 − 0.780i)17-s + (3.61 − 3.84i)18-s + (−3.64 + 2.10i)19-s + ⋯
L(s)  = 1  + (0.322 − 1.20i)2-s + (0.969 + 0.243i)3-s + (−0.474 − 0.273i)4-s + (0.199 − 0.979i)5-s + (0.604 − 1.08i)6-s + (0.397 − 0.397i)8-s + (0.881 + 0.472i)9-s + (−1.11 − 0.554i)10-s + (−0.405 − 0.234i)11-s + (−0.393 − 0.381i)12-s + (1.25 + 1.25i)13-s + (0.431 − 0.902i)15-s + (−0.623 − 1.08i)16-s + (0.706 − 0.189i)17-s + (0.851 − 0.907i)18-s + (−0.836 + 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76502 - 2.17259i\)
\(L(\frac12)\) \(\approx\) \(1.76502 - 2.17259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.421i)T \)
5 \( 1 + (-0.445 + 2.19i)T \)
7 \( 1 \)
good2 \( 1 + (-0.455 + 1.69i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (1.34 + 0.776i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.50 - 4.50i)T + 13iT^{2} \)
17 \( 1 + (-2.91 + 0.780i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.64 - 2.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.13 + 1.37i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.97T + 29T^{2} \)
31 \( 1 + (2.89 - 5.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.68 - 0.450i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (2.09 + 2.09i)T + 43iT^{2} \)
47 \( 1 + (0.0131 - 0.0489i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.57 + 5.88i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.46 + 4.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.65 - 2.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.625 - 2.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.73iT - 71T^{2} \)
73 \( 1 + (-9.92 + 2.66i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.11 + 1.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.2 - 12.2i)T - 83iT^{2} \)
89 \( 1 + (-0.678 - 1.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−10.15901558096613036009855716667, −9.411313546244963282031531224585, −8.629598907946249095194540539055, −7.916102709772212160742215405297, −6.62514763649751540817542658171, −5.25461904372499061196888871492, −4.10421526166779998387245264688, −3.67308676714063038600826489133, −2.23496871798124781104350834158, −1.43329218073379298567644537348, 1.94972449812571024840092037113, 3.16022839312591172828550329798, 4.16853314150218952135994503175, 5.68871342481144249170425588452, 6.23128864670515161614006682684, 7.23877985218031251452885930628, 7.87607342844006881212749259261, 8.435019449308617360317775106925, 9.693665172296337489602383049298, 10.51824922125984770952583688744

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