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Properties

Label 2-70-5.3-c4-0-8
Degree $2$
Conductor $70$
Sign $-0.805 - 0.592i$
Analytic cond. $7.23589$
Root an. cond. $2.68996$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)2-s + (−10.1 − 10.1i)3-s − 8i·4-s + (2.02 − 24.9i)5-s + 40.6·6-s + (13.0 − 13.0i)7-s + (16 + 16i)8-s + 125. i·9-s + (45.7 + 53.8i)10-s − 182.·11-s + (−81.2 + 81.2i)12-s + (−9.84 − 9.84i)13-s + 52.3i·14-s + (−273. + 232. i)15-s − 64·16-s + (173. − 173. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−1.12 − 1.12i)3-s − 0.5i·4-s + (0.0808 − 0.996i)5-s + 1.12·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 1.54i·9-s + (0.457 + 0.538i)10-s − 1.50·11-s + (−0.564 + 0.564i)12-s + (−0.0582 − 0.0582i)13-s + 0.267i·14-s + (−1.21 + 1.03i)15-s − 0.250·16-s + (0.599 − 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(7.23589\)
Root analytic conductor: \(2.68996\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :2),\ -0.805 - 0.592i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0460882 + 0.140363i\)
\(L(\frac12)\) \(\approx\) \(0.0460882 + 0.140363i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 2i)T \)
5 \( 1 + (-2.02 + 24.9i)T \)
7 \( 1 + (-13.0 + 13.0i)T \)
good3 \( 1 + (10.1 + 10.1i)T + 81iT^{2} \)
11 \( 1 + 182.T + 1.46e4T^{2} \)
13 \( 1 + (9.84 + 9.84i)T + 2.85e4iT^{2} \)
17 \( 1 + (-173. + 173. i)T - 8.35e4iT^{2} \)
19 \( 1 - 660. iT - 1.30e5T^{2} \)
23 \( 1 + (-22.6 - 22.6i)T + 2.79e5iT^{2} \)
29 \( 1 - 594. iT - 7.07e5T^{2} \)
31 \( 1 + 371.T + 9.23e5T^{2} \)
37 \( 1 + (-660. + 660. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.35e3T + 2.82e6T^{2} \)
43 \( 1 + (518. + 518. i)T + 3.41e6iT^{2} \)
47 \( 1 + (879. - 879. i)T - 4.87e6iT^{2} \)
53 \( 1 + (3.39e3 + 3.39e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 6.91e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.33e3T + 1.38e7T^{2} \)
67 \( 1 + (3.94e3 - 3.94e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 8.32e3T + 2.54e7T^{2} \)
73 \( 1 + (-7.07e3 - 7.07e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 7.27e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.71e3 + 7.71e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 435. iT - 6.27e7T^{2} \)
97 \( 1 + (-101. + 101. i)T - 8.85e7iT^{2} \)
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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

−13.01273907679458548989158213655, −12.37064220104585385206048204694, −11.11467607899455774220336570518, −9.915536028566272570269994212602, −8.141202936335398215556977000512, −7.47024490042233167090405240339, −5.88723358266430438175043314630, −5.10725833596427402409091913601, −1.50141695627252554472991544910, −0.10748176460699461886247553801, 2.85262778083263687600198689788, 4.65889996177325850858083565189, 5.99509875159372464484257822527, 7.64861234024360770179299547194, 9.386055882422281940109959954058, 10.49607602956986813151799066697, 10.90133636632733634811764266865, 11.89318440923999141969800433290, 13.34842738388600186666205547112, 15.07055186477087655034210930470

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