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Properties

Label 2-735-21.5-c1-0-45
Degree $2$
Conductor $735$
Sign $0.553 + 0.832i$
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  + (1.5 + 0.866i)2-s − 1.73i·3-s + (0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.49 − 2.59i)6-s − 1.73i·8-s − 2.99·9-s + (−1.5 + 0.866i)10-s + (3 − 1.73i)11-s + (1.50 − 0.866i)12-s − 3.46i·13-s + (1.49 + 0.866i)15-s + (2.49 − 4.33i)16-s + (−3 − 5.19i)17-s + (−4.49 − 2.59i)18-s + (6 + 3.46i)19-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s − 0.999i·3-s + (0.250 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.612 − 1.06i)6-s − 0.612i·8-s − 0.999·9-s + (−0.474 + 0.273i)10-s + (0.904 − 0.522i)11-s + (0.433 − 0.249i)12-s − 0.960i·13-s + (0.387 + 0.223i)15-s + (0.624 − 1.08i)16-s + (−0.727 − 1.26i)17-s + (−1.06 − 0.612i)18-s + (1.37 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09635 - 1.12332i\)
\(L(\frac12)\) \(\approx\) \(2.09635 - 1.12332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 0.866i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.73iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.30986572864683806260238432647, −9.283637333966890346433165406900, −8.167881731687850840018346598394, −7.31000313229571340733050202628, −6.66726941562789465227453115616, −5.85350390482400839288735701948, −5.07674130974125754058947313811, −3.70154989871732908996413724501, −2.83986354019931788790018960342, −0.933256382998182834583304606757, 1.93665306687718798642110202687, 3.33087541915833949707715243932, 4.15023812623342437303664274163, 4.67849208580994617409657087510, 5.65731500872816156413573041666, 6.74916628687139699289516607225, 8.179613566283504096843216732484, 9.004102997355472099512349656981, 9.687879286981908280421276478690, 10.79253791192828699264227887883

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