↑

Properties

Label 2-735-105.53-c1-0-62
Degree $2$
Conductor $735$
Sign $-0.520 - 0.853i$
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  + (0.517 − 1.93i)2-s + (−1.62 + 0.599i)3-s + (−1.73 − 0.999i)4-s + (−0.917 − 2.03i)5-s + (0.317 + 3.44i)6-s + (2.28 − 1.94i)9-s + (−4.41 + 0.717i)10-s + (−0.949 − 0.548i)11-s + (3.41 + 0.585i)12-s + (−2.43 − 2.43i)13-s + (2.71 + 2.76i)15-s + (−1.99 − 3.46i)16-s + (3.65 − 0.978i)17-s + (−2.58 − 5.41i)18-s + (−6.67 + 3.85i)19-s + (−0.449 + 4.44i)20-s + ⋯
L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.938 + 0.346i)3-s + (−0.866 − 0.499i)4-s + (−0.410 − 0.911i)5-s + (0.129 + 1.40i)6-s + (0.760 − 0.649i)9-s + (−1.39 + 0.226i)10-s + (−0.286 − 0.165i)11-s + (0.985 + 0.169i)12-s + (−0.676 − 0.676i)13-s + (0.700 + 0.713i)15-s + (−0.499 − 0.866i)16-s + (0.886 − 0.237i)17-s + (−0.609 − 1.27i)18-s + (−1.53 + 0.884i)19-s + (−0.100 + 0.994i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.520 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310065 + 0.552021i\)
\(L(\frac12)\) \(\approx\) \(0.310065 + 0.552021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.599i)T \)
5 \( 1 + (0.917 + 2.03i)T \)
7 \( 1 \)
good2 \( 1 + (-0.517 + 1.93i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (0.949 + 0.548i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.43 + 2.43i)T + 13iT^{2} \)
17 \( 1 + (-3.65 + 0.978i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.67 - 3.85i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.29 - 1.15i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 8.66T + 29T^{2} \)
31 \( 1 + (-1.73 + 3i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.71 + 1.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.89iT - 41T^{2} \)
43 \( 1 + (-1.89 - 1.89i)T + 43iT^{2} \)
47 \( 1 + (0.978 - 3.65i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.750 + 2.80i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.24 + 7.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.567 + 2.11i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 + (-9.65 + 2.58i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (10.3 - 5.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.34 + 5.34i)T - 83iT^{2} \)
89 \( 1 + (8.44 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.43 - 2.43i)T - 97iT^{2} \)
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

−9.970991272071492477622694729265, −9.509980677101976963246799114994, −8.205855733420103211220218401812, −7.22405799525537828750261708868, −5.77642470305837387364844738840, −5.01411689321317710788291928661, −4.19814980095543439099851969434, −3.28226557118273264288529195142, −1.65945471514462744895776353116, −0.32020244324562390041500069064, 2.21116627772797288072025415329, 4.01310004186215867038958741838, 4.95038972717436288688856830203, 5.79262817269252526777095097766, 6.77386070965092728897357244899, 7.08679298097863230753877956559, 7.85586095598288392158075455505, 8.972403523637338291068082956049, 10.39398835518589284097984000275, 10.86109731027385333167833515087

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