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Properties

Label 2-70-35.34-c2-0-1
Degree $2$
Conductor $70$
Sign $-0.991 - 0.128i$
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  + 1.41i·2-s − 2.07·3-s − 2.00·4-s + (−4.65 + 1.82i)5-s − 2.93i·6-s + (−3.36 + 6.13i)7-s − 2.82i·8-s − 4.67·9-s + (−2.57 − 6.58i)10-s + 1.67·11-s + 4.15·12-s + 9.81·13-s + (−8.67 − 4.76i)14-s + (9.67 − 3.78i)15-s + 4.00·16-s − 0.498·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.692·3-s − 0.500·4-s + (−0.931 + 0.364i)5-s − 0.489i·6-s + (−0.481 + 0.876i)7-s − 0.353i·8-s − 0.519·9-s + (−0.257 − 0.658i)10-s + 0.152·11-s + 0.346·12-s + 0.754·13-s + (−0.619 − 0.340i)14-s + (0.645 − 0.252i)15-s + 0.250·16-s − 0.0293·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.991 - 0.128i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ -0.991 - 0.128i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0305055 + 0.473101i\)
\(L(\frac12)\) \(\approx\) \(0.0305055 + 0.473101i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (4.65 - 1.82i)T \)
7 \( 1 + (3.36 - 6.13i)T \)
good3 \( 1 + 2.07T + 9T^{2} \)
11 \( 1 - 1.67T + 121T^{2} \)
13 \( 1 - 9.81T + 169T^{2} \)
17 \( 1 + 0.498T + 289T^{2} \)
19 \( 1 - 32.2iT - 361T^{2} \)
23 \( 1 - 3.78iT - 529T^{2} \)
29 \( 1 + 39.0T + 841T^{2} \)
31 \( 1 - 24.9iT - 961T^{2} \)
37 \( 1 + 16.0iT - 1.36e3T^{2} \)
41 \( 1 + 71.1iT - 1.68e3T^{2} \)
43 \( 1 - 37.7iT - 1.84e3T^{2} \)
47 \( 1 + 71.4T + 2.20e3T^{2} \)
53 \( 1 - 83.0iT - 2.80e3T^{2} \)
59 \( 1 - 36.4iT - 3.48e3T^{2} \)
61 \( 1 - 50.4iT - 3.72e3T^{2} \)
67 \( 1 + 12.2iT - 4.48e3T^{2} \)
71 \( 1 - 44.6T + 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 - 132.T + 6.24e3T^{2} \)
83 \( 1 - 8.72T + 6.88e3T^{2} \)
89 \( 1 + 54.0iT - 7.92e3T^{2} \)
97 \( 1 + 41.7T + 9.40e3T^{2} \)
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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

−15.13752877225914962112423531043, −14.19409724675582673029123190405, −12.60456124721659398979349043320, −11.78539871021973188299540466546, −10.65312736582807686046071578970, −9.023991489458987259750513389613, −7.924267463983326851262190665187, −6.44978936511238978820776075387, −5.51283121854205700867132086809, −3.61085001953339776493148495549, 0.45643559504228585344925247795, 3.50464024790795373558436113398, 4.88316582874211583142573595462, 6.65060725857433392915686960025, 8.203446663688495962037600516020, 9.510392731106526611662341385302, 11.13628148697086398662170611424, 11.32371285491388538492622364235, 12.74037278883248564922673742434, 13.56125864356641321540584165019

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