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Properties

Label 2-70-35.33-c1-0-2
Degree $2$
Conductor $70$
Sign $0.701 + 0.712i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (1.13 + 0.304i)3-s + (−0.866 + 0.499i)4-s + (0.264 − 2.22i)5-s − 1.17i·6-s + (0.698 + 2.55i)7-s + (0.707 + 0.707i)8-s + (−1.40 − 0.810i)9-s + (−2.21 + 0.318i)10-s + (−0.371 − 0.643i)11-s + (−1.13 + 0.304i)12-s + (−2.05 + 2.05i)13-s + (2.28 − 1.33i)14-s + (0.975 − 2.43i)15-s + (0.500 − 0.866i)16-s + (−1.69 + 6.33i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.655 + 0.175i)3-s + (−0.433 + 0.249i)4-s + (0.118 − 0.992i)5-s − 0.479i·6-s + (0.264 + 0.964i)7-s + (0.249 + 0.249i)8-s + (−0.467 − 0.270i)9-s + (−0.699 + 0.100i)10-s + (−0.112 − 0.194i)11-s + (−0.327 + 0.0877i)12-s + (−0.570 + 0.570i)13-s + (0.610 − 0.356i)14-s + (0.251 − 0.629i)15-s + (0.125 − 0.216i)16-s + (−0.411 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.878153 - 0.367939i\)
\(L(\frac12)\) \(\approx\) \(0.878153 - 0.367939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.264 + 2.22i)T \)
7 \( 1 + (-0.698 - 2.55i)T \)
good3 \( 1 + (-1.13 - 0.304i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.371 + 0.643i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.05 - 2.05i)T - 13iT^{2} \)
17 \( 1 + (1.69 - 6.33i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.946 - 1.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.11 + 1.36i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + (-2.96 + 1.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.691 + 2.58i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.817iT - 41T^{2} \)
43 \( 1 + (-1.59 - 1.59i)T + 43iT^{2} \)
47 \( 1 + (-4.54 + 1.21i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.29 + 4.81i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.27 - 2.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.25 - 3.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.2 - 3.54i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + (8.54 + 2.29i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.70 - 3.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.23 - 9.23i)T - 83iT^{2} \)
89 \( 1 + (3.01 - 5.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−14.56630668543515330480023651870, −13.30417103643065450337170925740, −12.34652524368064832044698451258, −11.42717765533083018224586568413, −9.814282040254112810080742509133, −8.817166168762131887747102067897, −8.248019391669049035146583107721, −5.81079438533627242104452397386, −4.21428472852267413872647120275, −2.29746218569794590901979806536, 2.97308106270516621375207486330, 5.05035891604859138792507413838, 6.96540857729424145917164614907, 7.54878922474035914957356947368, 8.968205322368970625169397565560, 10.27711069170329019053333743399, 11.27522386808903770358889681635, 13.18893465995032424232042940266, 14.06926692260383388252852662288, 14.65396263365242418635968285873

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