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Properties

Label 2-735-105.32-c1-0-65
Degree $2$
Conductor $735$
Sign $-0.437 + 0.899i$
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  + (0.355 + 0.0952i)2-s + (1.47 − 0.904i)3-s + (−1.61 − 0.932i)4-s + (1.32 − 1.79i)5-s + (0.611 − 0.180i)6-s + (−1.00 − 1.00i)8-s + (1.36 − 2.67i)9-s + (0.643 − 0.513i)10-s + (−2.93 − 1.69i)11-s + (−3.22 + 0.0835i)12-s + (1.59 − 1.59i)13-s + (0.331 − 3.85i)15-s + (1.60 + 2.77i)16-s + (−0.0514 − 0.192i)17-s + (0.739 − 0.820i)18-s + (−6.36 + 3.67i)19-s + ⋯
L(s)  = 1  + (0.251 + 0.0673i)2-s + (0.852 − 0.522i)3-s + (−0.807 − 0.466i)4-s + (0.593 − 0.804i)5-s + (0.249 − 0.0738i)6-s + (−0.355 − 0.355i)8-s + (0.454 − 0.890i)9-s + (0.203 − 0.162i)10-s + (−0.884 − 0.510i)11-s + (−0.931 + 0.0241i)12-s + (0.442 − 0.442i)13-s + (0.0856 − 0.996i)15-s + (0.400 + 0.694i)16-s + (−0.0124 − 0.0466i)17-s + (0.174 − 0.193i)18-s + (−1.45 + 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975173 - 1.55867i\)
\(L(\frac12)\) \(\approx\) \(0.975173 - 1.55867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 + 0.904i)T \)
5 \( 1 + (-1.32 + 1.79i)T \)
7 \( 1 \)
good2 \( 1 + (-0.355 - 0.0952i)T + (1.73 + i)T^{2} \)
11 \( 1 + (2.93 + 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.59 + 1.59i)T - 13iT^{2} \)
17 \( 1 + (0.0514 + 0.192i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.36 - 3.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.810 - 3.02i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 9.49T + 29T^{2} \)
31 \( 1 + (0.461 - 0.798i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.16 + 8.08i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 + 0.864i)T - 43iT^{2} \)
47 \( 1 + (-0.889 - 0.238i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.93 + 2.39i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.12 - 5.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.916 + 1.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.11 + 0.298i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 + (1.75 + 6.56i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.95 - 1.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.26 - 6.26i)T + 83iT^{2} \)
89 \( 1 + (-6.18 - 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.71 + 6.71i)T + 97iT^{2} \)
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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

−10.03770385193050970150666917540, −9.002306398180873040909394912841, −8.536215556173148135157137355189, −7.81469813163007384419615721022, −6.31178080961412188842258212361, −5.70019898374885061578290427039, −4.60985026145240298734108579816, −3.59362197256014623062749441740, −2.18106339362443347758557536874, −0.813537312472703208941633719299, 2.31839616237551239111014058331, 3.03181457850265218597636029689, 4.26929725767926491685266135824, 4.88562343008934960090580018211, 6.25106095636618944051104615660, 7.28971029682070676572691372588, 8.361018380711926093220883564536, 8.825320533571389072795086851473, 9.924581225044853498834358979655, 10.34444915069682356755750327509

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