Properties

Label 2-735-7.2-c1-0-3
Degree $2$
Conductor $735$
Sign $-0.386 - 0.922i$
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  + (−0.366 + 0.633i)2-s + (−0.5 − 0.866i)3-s + (0.732 + 1.26i)4-s + (0.5 − 0.866i)5-s + 0.732·6-s − 2.53·8-s + (−0.499 + 0.866i)9-s + (0.366 + 0.633i)10-s + (1.36 + 2.36i)11-s + (0.732 − 1.26i)12-s − 5.73·13-s − 0.999·15-s + (−0.535 + 0.928i)16-s + (3.36 + 5.83i)17-s + (−0.366 − 0.633i)18-s + (−1.23 + 2.13i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.448i)2-s + (−0.288 − 0.499i)3-s + (0.366 + 0.633i)4-s + (0.223 − 0.387i)5-s + 0.298·6-s − 0.896·8-s + (−0.166 + 0.288i)9-s + (0.115 + 0.200i)10-s + (0.411 + 0.713i)11-s + (0.211 − 0.366i)12-s − 1.58·13-s − 0.258·15-s + (−0.133 + 0.232i)16-s + (0.816 + 1.41i)17-s + (−0.0862 − 0.149i)18-s + (−0.282 + 0.489i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561066 + 0.843478i\)
\(L(\frac12)\) \(\approx\) \(0.561066 + 0.843478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.366 - 0.633i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.73T + 13T^{2} \)
17 \( 1 + (-3.36 - 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.23 - 2.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.633 + 1.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (-3.23 - 5.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.59 - 6.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.73T + 41T^{2} \)
43 \( 1 + 7.19T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.19 - 7.26i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.09 - 8.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.33 + 2.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.19T + 71T^{2} \)
73 \( 1 + (2.33 + 4.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.69 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + (4.56 - 7.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.07T + 97T^{2} \)
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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

−10.44668475019042233409137032774, −9.840579074815390818761906790068, −8.603923834807982323163367721286, −8.074919716569907113582633073196, −7.08805484137193944006437138104, −6.53567334321673434683030051952, −5.45047405511380910599690770327, −4.35431459642164100505704132356, −2.93093677127986404445460401394, −1.67477989739507423860542562299, 0.57055267335360952873773991915, 2.35302032581317506408621592450, 3.23320708775288127044705645235, 4.81589033667382310491896729310, 5.54465980339993046858210262925, 6.56811752403602120406425146070, 7.33341514600720588247743350159, 8.721372250765560766100448429187, 9.708973876764694101394155476029, 9.962853813876190852916624158394

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