Properties

Label 2-735-5.4-c1-0-13
Degree $2$
Conductor $735$
Sign $0.241 - 0.970i$
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  + 2.70i·2-s i·3-s − 5.34·4-s + (−2.17 − 0.539i)5-s + 2.70·6-s − 9.04i·8-s − 9-s + (1.46 − 5.87i)10-s + 2·11-s + 5.34i·12-s − 0.921i·13-s + (−0.539 + 2.17i)15-s + 13.8·16-s + 1.07i·17-s − 2.70i·18-s + 3.07·19-s + ⋯
L(s)  = 1  + 1.91i·2-s − 0.577i·3-s − 2.67·4-s + (−0.970 − 0.241i)5-s + 1.10·6-s − 3.19i·8-s − 0.333·9-s + (0.461 − 1.85i)10-s + 0.603·11-s + 1.54i·12-s − 0.255i·13-s + (−0.139 + 0.560i)15-s + 3.45·16-s + 0.261i·17-s − 0.638i·18-s + 0.706·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786576 + 0.615055i\)
\(L(\frac12)\) \(\approx\) \(0.786576 + 0.615055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (2.17 + 0.539i)T \)
7 \( 1 \)
good2 \( 1 - 2.70iT - 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 0.921iT - 13T^{2} \)
17 \( 1 - 1.07iT - 17T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 - 2.34iT - 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 + 6.52iT - 43T^{2} \)
47 \( 1 - 4.68iT - 47T^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 4.15T + 61T^{2} \)
67 \( 1 + 4.68iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 - 8.34T + 89T^{2} \)
97 \( 1 + 8.43iT - 97T^{2} \)
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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

−10.26821213639739374128150277543, −9.145830152758756534009856132248, −8.498803221593470222328730425214, −7.78800605821775798492419625957, −7.14015535641880697825137955338, −6.37939328279627589029127799178, −5.41850729321703544272927685129, −4.48649515361515054101412048689, −3.48850403083860805689671673557, −0.76204998513013853300185557497, 0.974767415738005954941835462056, 2.69326002907762983036432274868, 3.46127311182856594064911783111, 4.37110249655078203849110105381, 4.99884007345760852462677783950, 6.67120739198594822932706167834, 8.232129108213285429496995660977, 8.663562194962098141315196185590, 9.856318042085538302072425810107, 10.15870928020206685156924183249

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