↑

Properties

Label 2-35-35.19-c2-0-0
Degree $2$
Conductor $35$
Sign $0.999 - 0.0415i$
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  + (−2.85 − 1.65i)2-s + (2.20 + 3.81i)3-s + (3.44 + 5.97i)4-s + (3.20 − 3.83i)5-s − 14.5i·6-s + (5.32 + 4.54i)7-s − 9.55i·8-s + (−5.18 + 8.98i)9-s + (−15.4 + 5.68i)10-s + (−0.240 − 0.415i)11-s + (−15.1 + 26.2i)12-s − 11.0·13-s + (−7.70 − 21.7i)14-s + (21.6 + 3.76i)15-s + (−1.98 + 3.42i)16-s + (−9.10 − 15.7i)17-s + ⋯
L(s)  = 1  + (−1.42 − 0.825i)2-s + (0.733 + 1.27i)3-s + (0.861 + 1.49i)4-s + (0.640 − 0.767i)5-s − 2.42i·6-s + (0.760 + 0.649i)7-s − 1.19i·8-s + (−0.576 + 0.998i)9-s + (−1.54 + 0.568i)10-s + (−0.0218 − 0.0377i)11-s + (−1.26 + 2.19i)12-s − 0.853·13-s + (−0.550 − 1.55i)14-s + (1.44 + 0.250i)15-s + (−0.123 + 0.214i)16-s + (−0.535 − 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0415i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.999 - 0.0415i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.999 - 0.0415i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.735870 + 0.0153104i\)
\(L(\frac12)\) \(\approx\) \(0.735870 + 0.0153104i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.20 + 3.83i)T \)
7 \( 1 + (-5.32 - 4.54i)T \)
good2 \( 1 + (2.85 + 1.65i)T + (2 + 3.46i)T^{2} \)
3 \( 1 + (-2.20 - 3.81i)T + (-4.5 + 7.79i)T^{2} \)
11 \( 1 + (0.240 + 0.415i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 11.0T + 169T^{2} \)
17 \( 1 + (9.10 + 15.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (5.12 + 2.95i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (8.07 + 4.66i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 10.3T + 841T^{2} \)
31 \( 1 + (1.63 - 0.945i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-42.5 - 24.5i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 11.9iT - 1.68e3T^{2} \)
43 \( 1 + 62.7iT - 1.84e3T^{2} \)
47 \( 1 + (23.8 - 41.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (43.0 - 24.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (48.0 - 27.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (52.5 + 30.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-71.3 + 41.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 92.6T + 5.04e3T^{2} \)
73 \( 1 + (-24.5 - 42.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-1.72 + 2.98i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 30.5T + 6.88e3T^{2} \)
89 \( 1 + (-121. - 69.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 130.T + 9.40e3T^{2} \)
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

−16.61281961302496118689614292184, −15.44746094563261020472311903148, −14.13332576738908271148368914972, −12.25348374508173722728231704140, −10.93182928666085408671379034657, −9.671792042477010880196801019696, −9.137722380281251706833667377142, −8.113424785214548981584864935242, −4.85828602478504372609466522295, −2.41358545490348250493713385992, 1.85528495621075618483675962133, 6.38046613850693340600757126850, 7.40620939071695365998159556121, 8.194143535097466391092161833412, 9.680249641844707256909507813455, 10.95614990917892010232800166998, 13.00748057135426594521569338890, 14.29564924794087139792196544470, 14.98095277044562427030218168319, 16.85996072988256986503826888245

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