↑

Properties

Label 2-735-105.104-c1-0-21
Degree $2$
Conductor $735$
Sign $0.909 + 0.415i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.346·2-s + (−1.43 − 0.971i)3-s − 1.87·4-s + (−1.85 + 1.24i)5-s + (−0.496 − 0.336i)6-s − 1.34·8-s + (1.11 + 2.78i)9-s + (−0.644 + 0.430i)10-s − 0.537i·11-s + (2.69 + 1.82i)12-s − 3.81·13-s + (3.87 + 0.0267i)15-s + 3.29·16-s − 3.87i·17-s + (0.384 + 0.965i)18-s + 3.11i·19-s + ⋯
L(s)  = 1  + 0.244·2-s + (−0.827 − 0.561i)3-s − 0.939·4-s + (−0.831 + 0.555i)5-s + (−0.202 − 0.137i)6-s − 0.475·8-s + (0.370 + 0.928i)9-s + (−0.203 + 0.136i)10-s − 0.162i·11-s + (0.778 + 0.527i)12-s − 1.05·13-s + (0.999 + 0.00691i)15-s + 0.823·16-s − 0.940i·17-s + (0.0906 + 0.227i)18-s + 0.715i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.909 + 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670846 - 0.145929i\)
\(L(\frac12)\) \(\approx\) \(0.670846 - 0.145929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.43 + 0.971i)T \)
5 \( 1 + (1.85 - 1.24i)T \)
7 \( 1 \)
good2 \( 1 - 0.346T + 2T^{2} \)
11 \( 1 + 0.537iT - 11T^{2} \)
13 \( 1 + 3.81T + 13T^{2} \)
17 \( 1 + 3.87iT - 17T^{2} \)
19 \( 1 - 3.11iT - 19T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 - 8.41iT - 29T^{2} \)
31 \( 1 + 3.02iT - 31T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 - 8.56T + 41T^{2} \)
43 \( 1 + 4.12iT - 43T^{2} \)
47 \( 1 - 0.416iT - 47T^{2} \)
53 \( 1 - 4.28T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 0.282iT - 61T^{2} \)
67 \( 1 - 9.68iT - 67T^{2} \)
71 \( 1 - 1.01iT - 71T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 - 6.16T + 89T^{2} \)
97 \( 1 - 8.00T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51943302434780844499217498274, −9.503300163732710948576164527803, −8.535738640240843358370967694581, −7.41271163979674343965883504969, −7.05623380458951138456032252895, −5.67138342979663854263268451825, −4.97730356900506779561270379369, −3.99699251852414515289578962424, −2.72627179764388119742794627524, −0.64080166616684553540964195544, 0.74795180610306666105467322689, 3.22014379649876553952076161234, 4.41328434773090010694337357018, 4.72862988930375904743953036565, 5.67824782966662006314523109897, 6.86483183575028332561552050652, 7.945167152188496216227699484902, 8.895346779575757398096115770309, 9.565050730542741044240898755537, 10.40394864372787230492802371659

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