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Properties

Label 2-735-7.4-c1-0-21
Degree $2$
Conductor $735$
Sign $-0.605 - 0.795i$
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  + (−1.11 − 1.93i)2-s + (0.5 − 0.866i)3-s + (−1.5 + 2.59i)4-s + (0.5 + 0.866i)5-s − 2.23·6-s + 2.23·8-s + (−0.499 − 0.866i)9-s + (1.11 − 1.93i)10-s + (1.23 − 2.14i)11-s + (1.50 + 2.59i)12-s − 4.47·13-s + 0.999·15-s + (0.499 + 0.866i)16-s + (1 − 1.73i)17-s + (−1.11 + 1.93i)18-s + (−3.23 − 5.60i)19-s + ⋯
L(s)  = 1  + (−0.790 − 1.36i)2-s + (0.288 − 0.499i)3-s + (−0.750 + 1.29i)4-s + (0.223 + 0.387i)5-s − 0.912·6-s + 0.790·8-s + (−0.166 − 0.288i)9-s + (0.353 − 0.612i)10-s + (0.372 − 0.645i)11-s + (0.433 + 0.749i)12-s − 1.24·13-s + 0.258·15-s + (0.124 + 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.263 + 0.456i)18-s + (−0.742 − 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237752 + 0.479602i\)
\(L(\frac12)\) \(\approx\) \(0.237752 + 0.479602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.11 + 1.93i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.23 + 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.23 + 5.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (5.23 - 9.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.47 + 9.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8.94T + 43T^{2} \)
47 \( 1 + (-2.47 - 4.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.23 + 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.47 - 7.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (-1.76 + 3.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.47 - 4.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.944T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−9.904065673761121605249884107176, −9.054158095291252054087391934897, −8.534364846864236732170055215138, −7.35507989455265822257232435521, −6.57717233494065938225670977925, −5.17141695414646806116174598558, −3.69236108011718957281815850222, −2.72787487584837948335430939110, −1.91464459574307210017226161144, −0.32659915608370331356314198235, 1.92521740391987320148288821105, 3.76406455234968437947791294648, 4.93682640775449739583981812286, 5.73072100796744867386152415230, 6.69944583373681377336154761971, 7.67058314773008364313889522775, 8.218648032064520578563075607625, 9.184753909315392548937122914065, 9.809507850673465420052567399260, 10.28748356085444178849188995109

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