Properties

Label 2-735-7.4-c1-0-12
Degree $2$
Conductor $735$
Sign $0.701 - 0.712i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + 3·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (2 − 3.46i)11-s + (0.499 + 0.866i)12-s + 2·13-s − 0.999·15-s + (0.500 + 0.866i)16-s + (1 − 1.73i)17-s + (0.499 − 0.866i)18-s + (2 + 3.46i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.603 − 1.04i)11-s + (0.144 + 0.249i)12-s + 0.554·13-s − 0.258·15-s + (0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (0.117 − 0.204i)18-s + (0.458 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.701 - 0.712i$
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: \((2,\ 735,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92754 + 0.807802i\)
\(L(\frac12)\) \(\approx\) \(1.92754 + 0.807802i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.44085262186650594455403658308, −9.820753679276231579308735629264, −8.780385428561799023986680720238, −7.74200323337259736412749113934, −6.68514549546885768185179291607, −6.00341736346239346928464355066, −5.37134826755471197276567573114, −4.19108861556463827089047570082, −3.10377864562180349116400859595, −1.31522257443520703242771185894, 1.36853514456021684208502305884, 2.38798619110382202046358205850, 3.73662548265545736723540908588, 4.64132482839222852623461601140, 5.76200086835396758228929546805, 6.90023620476517586462807351611, 7.50744582753278594137804903361, 8.559098329660752109538996957360, 9.493012922904278561763299035601, 10.52116086848308035956223612003

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