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Properties

Label 2-70-35.2-c2-0-4
Degree $2$
Conductor $70$
Sign $0.907 + 0.420i$
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  + (1.36 − 0.366i)2-s + (−1.58 − 0.423i)3-s + (1.73 − i)4-s + (4.94 + 0.755i)5-s − 2.31·6-s + (6.42 − 2.78i)7-s + (1.99 − 2i)8-s + (−5.47 − 3.15i)9-s + (7.02 − 0.776i)10-s + (−0.341 − 0.591i)11-s + (−3.16 + 0.847i)12-s + (−11.9 + 11.9i)13-s + (7.75 − 6.15i)14-s + (−7.5 − 3.29i)15-s + (1.99 − 3.46i)16-s + (−8.70 + 32.4i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.527 − 0.141i)3-s + (0.433 − 0.250i)4-s + (0.988 + 0.151i)5-s − 0.386·6-s + (0.917 − 0.398i)7-s + (0.249 − 0.250i)8-s + (−0.607 − 0.350i)9-s + (0.702 − 0.0776i)10-s + (−0.0310 − 0.0538i)11-s + (−0.263 + 0.0706i)12-s + (−0.919 + 0.919i)13-s + (0.553 − 0.439i)14-s + (−0.5 − 0.219i)15-s + (0.124 − 0.216i)16-s + (−0.512 + 1.91i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.907 + 0.420i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.66731 - 0.367778i\)
\(L(\frac12)\) \(\approx\) \(1.66731 - 0.367778i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 + (-4.94 - 0.755i)T \)
7 \( 1 + (-6.42 + 2.78i)T \)
good3 \( 1 + (1.58 + 0.423i)T + (7.79 + 4.5i)T^{2} \)
11 \( 1 + (0.341 + 0.591i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (11.9 - 11.9i)T - 169iT^{2} \)
17 \( 1 + (8.70 - 32.4i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (8.34 + 4.81i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (6.26 + 23.3i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 16.5iT - 841T^{2} \)
31 \( 1 + (11.9 + 20.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (25.3 - 6.79i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 0.200T + 1.68e3T^{2} \)
43 \( 1 + (41.1 - 41.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-46.1 + 12.3i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-90.1 - 24.1i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-20.0 + 11.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.08 + 10.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-0.363 + 1.35i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 63.2T + 5.04e3T^{2} \)
73 \( 1 + (45.5 + 12.1i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (97.2 + 56.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-23.1 + 23.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (69.1 + 39.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (50.6 + 50.6i)T + 9.40e3iT^{2} \)
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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

−14.43088416918124612779874098550, −13.31981591337311365310056296273, −12.20913192040001322169991552933, −11.14049817544918653535260526197, −10.24323208149969880207753072152, −8.628132089424817896962943543284, −6.78399104646826823587783721630, −5.78273199385184733486754294363, −4.40448094961484560822633007321, −2.06038968024170684479780954854, 2.47301970310015513835833818163, 5.08980419575289711622813427084, 5.47392389271303108327165994693, 7.20199633118866867844809447315, 8.740216819734144704025528918076, 10.24079466563238677280049993581, 11.39034353743832150085906133659, 12.32165913250138333155972091951, 13.64769549656181631438175902470, 14.35090697918093222321238700682

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