Properties

Label 2-735-105.32-c1-0-40
Degree $2$
Conductor $735$
Sign $0.524 + 0.851i$
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  + (−0.474 − 0.127i)2-s + (1.40 + 1.01i)3-s + (−1.52 − 0.879i)4-s + (−2.23 − 0.0580i)5-s + (−0.536 − 0.659i)6-s + (1.30 + 1.30i)8-s + (0.942 + 2.84i)9-s + (1.05 + 0.311i)10-s + (−2.31 − 1.33i)11-s + (−1.24 − 2.78i)12-s + (2.14 − 2.14i)13-s + (−3.07 − 2.34i)15-s + (1.30 + 2.26i)16-s + (−1.19 − 4.46i)17-s + (−0.0850 − 1.46i)18-s + (4.54 − 2.62i)19-s + ⋯
L(s)  = 1  + (−0.335 − 0.0898i)2-s + (0.810 + 0.585i)3-s + (−0.761 − 0.439i)4-s + (−0.999 − 0.0259i)5-s + (−0.219 − 0.269i)6-s + (0.461 + 0.461i)8-s + (0.314 + 0.949i)9-s + (0.332 + 0.0984i)10-s + (−0.697 − 0.402i)11-s + (−0.359 − 0.802i)12-s + (0.596 − 0.596i)13-s + (−0.795 − 0.606i)15-s + (0.326 + 0.565i)16-s + (−0.290 − 1.08i)17-s + (−0.0200 − 0.346i)18-s + (1.04 − 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.524 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879369 - 0.490983i\)
\(L(\frac12)\) \(\approx\) \(0.879369 - 0.490983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 - 1.01i)T \)
5 \( 1 + (2.23 + 0.0580i)T \)
7 \( 1 \)
good2 \( 1 + (0.474 + 0.127i)T + (1.73 + i)T^{2} \)
11 \( 1 + (2.31 + 1.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \)
17 \( 1 + (1.19 + 4.46i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.54 + 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.932 + 3.48i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.784 - 2.92i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \)
47 \( 1 + (10.4 + 2.80i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.05 + 1.62i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.0797 + 0.138i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.36 + 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.39 + 1.98i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (-1.52 - 5.68i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.37 + 1.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 + 4.03i)T + 83iT^{2} \)
89 \( 1 + (-1.97 - 3.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.86 + 1.86i)T + 97iT^{2} \)
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

−10.16310003464149489861564416001, −9.314856422353710373261146654498, −8.500789181591754462869940517392, −8.062473318888883635858898317277, −7.07105261352349229388904337302, −5.37295503634513381867934939242, −4.73464424381397044465669549370, −3.69495844723993375226522744891, −2.69776154215791167469560214568, −0.61994337522005329733696769819, 1.31980768830483643012138497947, 3.11448650094986540099243598683, 3.85841026077876126181677042960, 4.83965300733392742367580792856, 6.44053184067044227544975200546, 7.44508397473915634045939846941, 8.002416604685368384494456511084, 8.576908184682602916360536180065, 9.425967597496623854399006384070, 10.30270306618055805807737915815

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