Properties

Label 2-735-105.23-c1-0-64
Degree $2$
Conductor $735$
Sign $-0.0610 + 0.998i$
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  + (2.45 − 0.658i)2-s + (−1.25 − 1.19i)3-s + (3.87 − 2.23i)4-s + (1.98 − 1.02i)5-s + (−3.87 − 2.10i)6-s + (4.45 − 4.45i)8-s + (0.154 + 2.99i)9-s + (4.21 − 3.82i)10-s + (1.35 − 0.784i)11-s + (−7.53 − 1.81i)12-s + (−2.21 − 2.21i)13-s + (−3.71 − 1.08i)15-s + (3.54 − 6.14i)16-s + (−1.32 + 4.92i)17-s + (2.35 + 7.26i)18-s + (1.45 + 0.840i)19-s + ⋯
L(s)  = 1  + (1.73 − 0.465i)2-s + (−0.725 − 0.688i)3-s + (1.93 − 1.11i)4-s + (0.889 − 0.457i)5-s + (−1.58 − 0.859i)6-s + (1.57 − 1.57i)8-s + (0.0514 + 0.998i)9-s + (1.33 − 1.20i)10-s + (0.409 − 0.236i)11-s + (−2.17 − 0.523i)12-s + (−0.615 − 0.615i)13-s + (−0.959 − 0.280i)15-s + (0.886 − 1.53i)16-s + (−0.320 + 1.19i)17-s + (0.554 + 1.71i)18-s + (0.333 + 0.192i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0610 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0610 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0610 + 0.998i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.0610 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57762 - 2.74006i\)
\(L(\frac12)\) \(\approx\) \(2.57762 - 2.74006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.25 + 1.19i)T \)
5 \( 1 + (-1.98 + 1.02i)T \)
7 \( 1 \)
good2 \( 1 + (-2.45 + 0.658i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-1.35 + 0.784i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.21 + 2.21i)T + 13iT^{2} \)
17 \( 1 + (1.32 - 4.92i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.45 - 0.840i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.364 - 1.36i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.91T + 29T^{2} \)
31 \( 1 + (-1.37 - 2.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.161 - 0.601i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 + 5.47i)T + 43iT^{2} \)
47 \( 1 + (-5.04 + 1.35i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.87 + 1.03i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (-5.12 - 1.37i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.61iT - 71T^{2} \)
73 \( 1 + (-2.15 + 8.05i)T + (-63.2 - 36.5i)T^{2} \)
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

−10.59333308566833016212166982621, −9.687524955606661908146488562648, −8.244786343998088718970066524996, −7.00884543582870016648063944608, −6.20975191487039894918467608166, −5.54085481870479184357174746645, −4.94598892565208984037476514397, −3.73548249779430901402857627923, −2.35287914014542050608069459323, −1.43146948153093657815699686271, 2.26758442134222646895366005480, 3.42087009787124002108927049059, 4.47004692614598622774349234557, 5.18649842269352495811529134204, 5.92153427850637254772689421428, 6.78222942419627199934765848834, 7.29442598024433360355942877294, 9.199680770748069759043465302608, 9.810930854359834437495470962733, 11.01319712170664327400839428180

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