↑

Properties

Label 2-35-7.3-c2-0-5
Degree $2$
Conductor $35$
Sign $-0.999 - 0.0406i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.77 − 3.08i)2-s + (−2.10 − 1.21i)3-s + (−4.32 + 7.49i)4-s + (1.93 − 1.11i)5-s + 8.65i·6-s + (−5.39 − 4.46i)7-s + 16.5·8-s + (−1.53 − 2.66i)9-s + (−6.88 − 3.97i)10-s + (6.95 − 12.0i)11-s + (18.2 − 10.5i)12-s + 0.702i·13-s + (−4.15 + 24.5i)14-s − 5.44·15-s + (−12.1 − 21.0i)16-s + (23.6 + 13.6i)17-s + ⋯
L(s)  = 1  + (−0.889 − 1.54i)2-s + (−0.702 − 0.405i)3-s + (−1.08 + 1.87i)4-s + (0.387 − 0.223i)5-s + 1.44i·6-s + (−0.770 − 0.637i)7-s + 2.06·8-s + (−0.171 − 0.296i)9-s + (−0.688 − 0.397i)10-s + (0.631 − 1.09i)11-s + (1.51 − 0.877i)12-s + 0.0540i·13-s + (−0.296 + 1.75i)14-s − 0.362·15-s + (−0.759 − 1.31i)16-s + (1.38 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0406i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.999 - 0.0406i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -0.999 - 0.0406i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00962513 + 0.472840i\)
\(L(\frac12)\) \(\approx\) \(0.00962513 + 0.472840i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (5.39 + 4.46i)T \)
good2 \( 1 + (1.77 + 3.08i)T + (-2 + 3.46i)T^{2} \)
3 \( 1 + (2.10 + 1.21i)T + (4.5 + 7.79i)T^{2} \)
11 \( 1 + (-6.95 + 12.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 0.702iT - 169T^{2} \)
17 \( 1 + (-23.6 - 13.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.21 + 1.27i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (16.4 + 28.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 3.39T + 841T^{2} \)
31 \( 1 + (-20.7 - 11.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.9 - 20.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 25.1iT - 1.68e3T^{2} \)
43 \( 1 + 25.1T + 1.84e3T^{2} \)
47 \( 1 + (5.59 - 3.23i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-18.0 + 31.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (32.0 + 18.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (30.3 - 17.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-22.8 + 39.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 80.4T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 44.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (0.0415 + 0.0720i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 95.3iT - 6.88e3T^{2} \)
89 \( 1 + (-104. + 60.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 0.362iT - 9.40e3T^{2} \)
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

−16.65268316151392164375984375700, −14.00632609687542781771943200977, −12.70489201633304790703929498500, −11.95349321623337891536206767297, −10.73152332191588766189448935723, −9.733749242576640471175109618033, −8.401613484469717258337617237536, −6.28568831326878156375543387186, −3.48534140172571030409630816687, −0.867534250729885271776125483445, 5.27035878004779938526391022854, 6.28721122871296951842842006790, 7.67611432823142226889968086880, 9.443853312760467619286778714391, 10.02720376547527547408962706174, 11.95172964242655148015676784097, 13.88370597495734095947165829034, 15.05689406628713517937791072408, 16.02582221410289693380124929211, 16.82095948522795932921717036572

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