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Properties

Label 2-70-35.32-c2-0-1
Degree $2$
Conductor $70$
Sign $0.663 - 0.747i$
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 + (1.35 + 5.05i)3-s + (−1.73 − i)4-s + (−3.48 + 3.58i)5-s + 7.39·6-s + (6.88 + 1.26i)7-s + (−2 + 1.99i)8-s + (−15.8 + 9.16i)9-s + (3.61 + 6.07i)10-s + (5.56 − 9.64i)11-s + (2.70 − 10.1i)12-s + (9.62 − 9.62i)13-s + (4.25 − 8.94i)14-s + (−22.8 − 12.7i)15-s + (1.99 + 3.46i)16-s + (8.55 − 2.29i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.451 + 1.68i)3-s + (−0.433 − 0.250i)4-s + (−0.697 + 0.716i)5-s + 1.23·6-s + (0.983 + 0.181i)7-s + (−0.250 + 0.249i)8-s + (−1.76 + 1.01i)9-s + (0.361 + 0.607i)10-s + (0.506 − 0.876i)11-s + (0.225 − 0.841i)12-s + (0.740 − 0.740i)13-s + (0.303 − 0.638i)14-s + (−1.52 − 0.851i)15-s + (0.124 + 0.216i)16-s + (0.503 − 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.663 - 0.747i$
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: \((2,\ 70,\ (\ :1),\ 0.663 - 0.747i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.28812 + 0.579122i\)
\(L(\frac12)\) \(\approx\) \(1.28812 + 0.579122i\)
\(L(2)\) 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 \( 1 + (3.48 - 3.58i)T \)
7 \( 1 + (-6.88 - 1.26i)T \)
good3 \( 1 + (-1.35 - 5.05i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (-5.56 + 9.64i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-9.62 + 9.62i)T - 169iT^{2} \)
17 \( 1 + (-8.55 + 2.29i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-5.79 + 3.34i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (27.5 + 7.37i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 29.0iT - 841T^{2} \)
31 \( 1 + (11.9 - 20.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-4.01 + 14.9i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 9.18T + 1.68e3T^{2} \)
43 \( 1 + (-55.1 + 55.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (2.29 - 8.55i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (3.69 + 13.8i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (67.7 + 39.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (10.7 + 18.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.3 - 13.7i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 101.T + 5.04e3T^{2} \)
73 \( 1 + (-27.1 - 101. i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-11.7 + 6.81i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-28.4 + 28.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (-6.56 + 3.78i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-74.4 - 74.4i)T + 9.40e3iT^{2} \)
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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

−14.48765420133033819549184467200, −14.05312191967097589239989393699, −11.96632607126560375503502769885, −10.93896173715451247037606807436, −10.50239853485068404773894282114, −9.018251514438437357190163671762, −8.094414132184972169802277492588, −5.53130674869938544638240815183, −4.09605239174944556951494984643, −3.15547939031342570732233677414, 1.49532513100505645029990022309, 4.22667808319286959996443441848, 6.08213373174498542732030416834, 7.51662153027755617319037076597, 7.966759457794217126543386800243, 9.125158257022509741785212289507, 11.64836609380154125804981041807, 12.22355498457420860436560915137, 13.37964842196653393447145232191, 14.16869148160008323518352563370

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