Properties

Label 2-70-35.3-c1-0-0
Degree $2$
Conductor $70$
Sign $0.509 - 0.860i$
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  + (−0.965 − 0.258i)2-s + (0.523 + 1.95i)3-s + (0.866 + 0.499i)4-s + (−2.03 + 0.935i)5-s − 2.02i·6-s + (1.83 + 1.90i)7-s + (−0.707 − 0.707i)8-s + (−0.941 + 0.543i)9-s + (2.20 − 0.378i)10-s + (2.01 − 3.49i)11-s + (−0.523 + 1.95i)12-s + (0.204 − 0.204i)13-s + (−1.28 − 2.31i)14-s + (−2.89 − 3.47i)15-s + (0.500 + 0.866i)16-s + (−1.97 + 0.527i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.302 + 1.12i)3-s + (0.433 + 0.249i)4-s + (−0.908 + 0.418i)5-s − 0.825i·6-s + (0.695 + 0.718i)7-s + (−0.249 − 0.249i)8-s + (−0.313 + 0.181i)9-s + (0.696 − 0.119i)10-s + (0.609 − 1.05i)11-s + (−0.151 + 0.563i)12-s + (0.0568 − 0.0568i)13-s + (−0.343 − 0.618i)14-s + (−0.746 − 0.897i)15-s + (0.125 + 0.216i)16-s + (−0.477 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.509 - 0.860i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.509 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622454 + 0.354602i\)
\(L(\frac12)\) \(\approx\) \(0.622454 + 0.354602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (2.03 - 0.935i)T \)
7 \( 1 + (-1.83 - 1.90i)T \)
good3 \( 1 + (-0.523 - 1.95i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.01 + 3.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.204 + 0.204i)T - 13iT^{2} \)
17 \( 1 + (1.97 - 0.527i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.10 + 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.17 + 4.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.15iT - 29T^{2} \)
31 \( 1 + (-6.33 - 3.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.46 + 1.19i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 + (4.97 + 4.97i)T + 43iT^{2} \)
47 \( 1 + (0.0815 - 0.304i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.00 - 2.14i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.427 + 0.740i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.99 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.817 + 3.05i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + (2.98 + 11.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.39 - 2.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.85 - 3.85i)T - 83iT^{2} \)
89 \( 1 + (1.53 + 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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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

−15.22918413582414997142984669262, −14.24219474580634956196800206242, −12.31274380923542050071877369425, −11.16181088725231799729047903862, −10.60151321426083269417982551579, −8.908677182057046311054470353093, −8.522977572268320191167320686536, −6.73844098056840480408485539956, −4.63057072172908045307350318820, −3.13904817475299052291053449971, 1.57751405592454151361540136058, 4.36288301637604859248062731712, 6.67808905485602647026870735832, 7.68324315817142882728372994284, 8.286034516612125050687061433734, 9.878522828395303252848506428670, 11.38988144769205107773740841200, 12.24209159715530163845724630915, 13.39635362248342307623637135421, 14.60890409366785953523866453667

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