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Properties

Label 2-70-35.32-c2-0-6
Degree $2$
Conductor $70$
Sign $-0.773 + 0.633i$
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  + (0.366 − 1.36i)2-s + (−1.31 − 4.90i)3-s + (−1.73 − i)4-s + (2.11 + 4.52i)5-s − 7.18·6-s + (−0.145 − 6.99i)7-s + (−2 + 1.99i)8-s + (−14.5 + 8.41i)9-s + (6.96 − 1.23i)10-s + (0.474 − 0.821i)11-s + (−2.63 + 9.81i)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 + (29.3 − 7.87i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.438 − 1.63i)3-s + (−0.433 − 0.250i)4-s + (0.423 + 0.905i)5-s − 1.19·6-s + (−0.0207 − 0.999i)7-s + (−0.250 + 0.249i)8-s + (−1.61 + 0.934i)9-s + (0.696 − 0.123i)10-s + (0.0431 − 0.0746i)11-s + (−0.219 + 0.818i)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 + (1.72 − 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.773 + 0.633i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ -0.773 + 0.633i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.389921 - 1.09119i\)
\(L(\frac12)\) \(\approx\) \(0.389921 - 1.09119i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + (-2.11 - 4.52i)T \)
7 \( 1 + (0.145 + 6.99i)T \)
good3 \( 1 + (1.31 + 4.90i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (-0.474 + 0.821i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.862 + 0.862i)T - 169iT^{2} \)
17 \( 1 + (-29.3 + 7.87i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-20.9 + 12.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (5.47 + 1.46i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 7.33iT - 841T^{2} \)
31 \( 1 + (23.5 - 40.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (2.13 - 7.95i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 53.3T + 1.68e3T^{2} \)
43 \( 1 + (33.0 - 33.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (7.87 - 29.3i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-3.48 - 12.9i)T + (-2.43e3 + 1.40e3i)T^{2} \)
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 \( 1 + (-65.9 + 17.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 11.9T + 5.04e3T^{2} \)
73 \( 1 + (14.2 + 53.2i)T + (-4.61e3 + 2.66e3i)T^{2} \)
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

−13.87450456859167726354047953036, −12.94276618269686396858540211281, −11.86207503074993714456566436998, −10.97022906637181794782066841342, −9.810233154624533027089308180391, −7.74117893483917522625530221186, −6.91462104124605693822960308069, −5.58422801427215488380720902768, −3.06367175391405846480066848591, −1.19697703192136126021798360370, 3.78575260906397450670996576677, 5.33480485511698825598632164310, 5.70594545092454461310455509358, 8.182816733098446546946094486683, 9.373138105025773366637163904949, 9.952636073457636130106308741034, 11.65081463581181786610089254755, 12.61599413529274006611151206256, 14.23656878877136704300166381365, 15.07496665943165804694992873531

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