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Properties

Label 2-735-105.53-c1-0-0
Degree $2$
Conductor $735$
Sign $0.999 + 0.0444i$
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  + (−0.658 + 2.45i)2-s + (−0.405 + 1.68i)3-s + (−3.87 − 2.23i)4-s + (−0.108 + 2.23i)5-s + (−3.87 − 2.10i)6-s + (4.45 − 4.45i)8-s + (−2.67 − 1.36i)9-s + (−5.41 − 1.73i)10-s + (−1.35 − 0.784i)11-s + (5.34 − 5.62i)12-s + (−2.21 − 2.21i)13-s + (−3.71 − 1.08i)15-s + (3.54 + 6.14i)16-s + (4.92 − 1.32i)17-s + (5.11 − 5.66i)18-s + (−1.45 + 0.840i)19-s + ⋯
L(s)  = 1  + (−0.465 + 1.73i)2-s + (−0.233 + 0.972i)3-s + (−1.93 − 1.11i)4-s + (−0.0485 + 0.998i)5-s + (−1.58 − 0.859i)6-s + (1.57 − 1.57i)8-s + (−0.890 − 0.454i)9-s + (−1.71 − 0.549i)10-s + (−0.409 − 0.236i)11-s + (1.54 − 1.62i)12-s + (−0.615 − 0.615i)13-s + (−0.959 − 0.280i)15-s + (0.886 + 1.53i)16-s + (1.19 − 0.320i)17-s + (1.20 − 1.33i)18-s + (−0.333 + 0.192i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.999 + 0.0444i$
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: \((2,\ 735,\ (\ :1/2),\ 0.999 + 0.0444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0617391 - 0.00137327i\)
\(L(\frac12)\) \(\approx\) \(0.0617391 - 0.00137327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.405 - 1.68i)T \)
5 \( 1 + (0.108 - 2.23i)T \)
7 \( 1 \)
good2 \( 1 + (0.658 - 2.45i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (1.35 + 0.784i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.21 + 2.21i)T + 13iT^{2} \)
17 \( 1 + (-4.92 + 1.32i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.45 - 0.840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.36 + 0.364i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.91T + 29T^{2} \)
31 \( 1 + (-1.37 + 2.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.601 + 0.161i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 + 5.47i)T + 43iT^{2} \)
47 \( 1 + (1.35 - 5.04i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.03 - 3.87i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.77 + 4.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.70 - 6.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.37 + 5.12i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.61iT - 71T^{2} \)
73 \( 1 + (8.05 - 2.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (14.7 - 8.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.21 - 3.21i)T - 83iT^{2} \)
89 \( 1 + (4.70 + 8.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.39 - 4.39i)T - 97iT^{2} \)
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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

−10.82599928901190414935012772898, −9.952168723796884809813480367633, −9.642128341647098277408204570544, −8.418726941974696945399078361842, −7.73269913804537786064116676732, −6.93834054140736617082571334913, −5.81117976974415096480098387113, −5.47458411674161940554065024794, −4.24324878614134209847338225018, −3.03760740435243901925372025434, 0.03962764276607227324987638778, 1.41616828693334068406242127648, 2.21947852560919442985915899693, 3.55524088237414219894603029117, 4.74567977449726351621270589106, 5.66992060627770637817473021917, 7.25692727664835353016674531265, 8.119132206990292015961181997097, 8.817791657343699979020786300912, 9.677188107127899583435582751659

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