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Properties

Label 2-70-35.18-c2-0-3
Degree $2$
Conductor $70$
Sign $0.969 + 0.245i$
Analytic cond. $1.90736$
Root an. cond. $1.38107$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (4.90 − 1.31i)3-s + (1.73 + i)4-s + (2.86 + 4.09i)5-s − 7.18·6-s + (−6.99 + 0.145i)7-s + (−1.99 − 2i)8-s + (14.5 − 8.41i)9-s + (−2.41 − 6.64i)10-s + (0.474 − 0.821i)11-s + (9.81 + 2.63i)12-s + (0.862 + 0.862i)13-s + (9.61 + 2.36i)14-s + (19.4 + 16.3i)15-s + (1.99 + 3.46i)16-s + (−7.87 − 29.3i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (1.63 − 0.438i)3-s + (0.433 + 0.250i)4-s + (0.572 + 0.819i)5-s − 1.19·6-s + (−0.999 + 0.0207i)7-s + (−0.249 − 0.250i)8-s + (1.61 − 0.934i)9-s + (−0.241 − 0.664i)10-s + (0.0431 − 0.0746i)11-s + (0.818 + 0.219i)12-s + (0.0663 + 0.0663i)13-s + (0.686 + 0.168i)14-s + (1.29 + 1.09i)15-s + (0.124 + 0.216i)16-s + (−0.463 − 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.969 + 0.245i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42231 - 0.177332i\)
\(L(\frac12)\) \(\approx\) \(1.42231 - 0.177332i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
5 \( 1 + (-2.86 - 4.09i)T \)
7 \( 1 + (6.99 - 0.145i)T \)
good3 \( 1 + (-4.90 + 1.31i)T + (7.79 - 4.5i)T^{2} \)
11 \( 1 + (-0.474 + 0.821i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.862 - 0.862i)T + 169iT^{2} \)
17 \( 1 + (7.87 + 29.3i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (20.9 - 12.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.46 + 5.47i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 7.33iT - 841T^{2} \)
31 \( 1 + (23.5 - 40.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-7.95 - 2.13i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 53.3T + 1.68e3T^{2} \)
43 \( 1 + (33.0 + 33.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (-29.3 - 7.87i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (12.9 - 3.48i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (43.5 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-22.7 - 39.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (17.6 + 65.9i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 11.9T + 5.04e3T^{2} \)
73 \( 1 + (-53.2 + 14.2i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-61.4 + 35.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-85.7 - 85.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-135. + 78.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (87.1 - 87.1i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.29776659717467816845530974326, −13.50916917351319309319134881815, −12.45407798790169573956768653380, −10.67594420002303701424314263184, −9.546914059772159157398455348711, −8.884404288329787593825309535983, −7.43025617834802589623578430434, −6.56047038840215223481132079219, −3.35084009219014865229468866165, −2.30107556862150294027785184457, 2.23058043713398732171991754249, 4.04290983837529895463820598660, 6.24515346848631143152287669632, 7.951564780155934398986612779487, 8.920201355310951771211896086499, 9.523159567859842214171236574170, 10.56543778938090917423711253794, 12.80849797415061814651163357539, 13.35575583362676358170283050119, 14.74835414293624582101730700213

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