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Properties

Label 2-735-35.34-c2-0-72
Degree $2$
Conductor $735$
Sign $-0.990 - 0.136i$
Analytic cond. $20.0272$
Root an. cond. $4.47518$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.98i·2-s + 1.73·3-s + 0.0722·4-s + (−4.19 + 2.72i)5-s − 3.43i·6-s − 8.07i·8-s + 2.99·9-s + (5.40 + 8.30i)10-s + 6.60·11-s + 0.125·12-s − 21.3·13-s + (−7.25 + 4.72i)15-s − 15.7·16-s − 15.9·17-s − 5.94i·18-s − 20.0i·19-s + ⋯
L(s)  = 1  − 0.990i·2-s + 0.577·3-s + 0.0180·4-s + (−0.838 + 0.545i)5-s − 0.572i·6-s − 1.00i·8-s + 0.333·9-s + (0.540 + 0.830i)10-s + 0.600·11-s + 0.0104·12-s − 1.64·13-s + (−0.483 + 0.314i)15-s − 0.981·16-s − 0.941·17-s − 0.330i·18-s − 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.990 - 0.136i$
Analytic conductor: \(20.0272\)
Root analytic conductor: \(4.47518\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1),\ -0.990 - 0.136i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.081394465\)
\(L(\frac12)\) \(\approx\) \(1.081394465\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 + (4.19 - 2.72i)T \)
7 \( 1 \)
good2 \( 1 + 1.98iT - 4T^{2} \)
11 \( 1 - 6.60T + 121T^{2} \)
13 \( 1 + 21.3T + 169T^{2} \)
17 \( 1 + 15.9T + 289T^{2} \)
19 \( 1 + 20.0iT - 361T^{2} \)
23 \( 1 - 1.83iT - 529T^{2} \)
29 \( 1 + 4.58T + 841T^{2} \)
31 \( 1 + 60.6iT - 961T^{2} \)
37 \( 1 + 10.8iT - 1.36e3T^{2} \)
41 \( 1 - 18.1iT - 1.68e3T^{2} \)
43 \( 1 + 19.8iT - 1.84e3T^{2} \)
47 \( 1 + 87.6T + 2.20e3T^{2} \)
53 \( 1 + 13.3iT - 2.80e3T^{2} \)
59 \( 1 - 3.82iT - 3.48e3T^{2} \)
61 \( 1 - 36.0iT - 3.72e3T^{2} \)
67 \( 1 + 28.8iT - 4.48e3T^{2} \)
71 \( 1 + 98.4T + 5.04e3T^{2} \)
73 \( 1 - 120.T + 5.32e3T^{2} \)
79 \( 1 + 34.4T + 6.24e3T^{2} \)
83 \( 1 - 70.9T + 6.88e3T^{2} \)
89 \( 1 - 133. iT - 7.92e3T^{2} \)
97 \( 1 - 119.T + 9.40e3T^{2} \)
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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

−9.755640175715897415327817489529, −9.215195175969605282235814606704, −7.955664230211913936929740468200, −7.15508206473240995442026238061, −6.53209528459348440693098167294, −4.68902744978137536492537036954, −3.89765401769196387709521395412, −2.83383684501795646157147350046, −2.13258769496702238324561715127, −0.31918631829956225845500683934, 1.80707525180512519347216211412, 3.18188874978543492154059955310, 4.47820351208106461078603274846, 5.17718118246662630965974861958, 6.53958831083330906848963323442, 7.20977576598640176896758633445, 7.967858836854870583560176929383, 8.629624905573117612006809983551, 9.449746396256522517007487525311, 10.54820171751684504278725949544

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