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Properties

Label 2-70-35.32-c2-0-7
Degree $2$
Conductor $70$
Sign $-0.748 + 0.663i$
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 + (−0.707 − 2.63i)3-s + (−1.73 − i)4-s + (−3.39 − 3.66i)5-s − 3.86·6-s + (−1.73 + 6.78i)7-s + (−2 + 1.99i)8-s + (1.33 − 0.768i)9-s + (−6.25 + 3.29i)10-s + (10.0 − 17.3i)11-s + (−1.41 + 5.27i)12-s + (2.21 − 2.21i)13-s + (8.63 + 4.84i)14-s + (−7.27 + 11.5i)15-s + (1.99 + 3.46i)16-s + (−11.9 + 3.19i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.235 − 0.879i)3-s + (−0.433 − 0.250i)4-s + (−0.679 − 0.733i)5-s − 0.643·6-s + (−0.247 + 0.968i)7-s + (−0.250 + 0.249i)8-s + (0.147 − 0.0853i)9-s + (−0.625 + 0.329i)10-s + (0.911 − 1.57i)11-s + (−0.117 + 0.439i)12-s + (0.170 − 0.170i)13-s + (0.616 + 0.346i)14-s + (−0.485 + 0.770i)15-s + (0.124 + 0.216i)16-s + (−0.701 + 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.748 + 0.663i$
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: \((2,\ 70,\ (\ :1),\ -0.748 + 0.663i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.378886 - 0.998224i\)
\(L(\frac12)\) \(\approx\) \(0.378886 - 0.998224i\)
\(L(2)\) 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 \( 1 + (3.39 + 3.66i)T \)
7 \( 1 + (1.73 - 6.78i)T \)
good3 \( 1 + (0.707 + 2.63i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (-10.0 + 17.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.21 + 2.21i)T - 169iT^{2} \)
17 \( 1 + (11.9 - 3.19i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-15.7 + 9.11i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-39.7 - 10.6i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 19.0iT - 841T^{2} \)
31 \( 1 + (3.14 - 5.43i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (9.82 - 36.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 16.3T + 1.68e3T^{2} \)
43 \( 1 + (4.26 - 4.26i)T - 1.84e3iT^{2} \)
47 \( 1 + (-3.19 + 11.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (13.8 + 51.7i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-82.3 - 47.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (4.22 + 7.32i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.9 - 2.94i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 94.8T + 5.04e3T^{2} \)
73 \( 1 + (2.11 + 7.87i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (15.1 - 8.72i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (63.0 - 63.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (-67.0 + 38.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-105. - 105. i)T + 9.40e3iT^{2} \)
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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

−13.52606501325534245801037839319, −12.85563886718711162750154275105, −11.80861717123404270943989193162, −11.28203393317417124632320582456, −9.227551609875707815892411974134, −8.475542357197016859299271101314, −6.71840383025369087427405587677, −5.29776398745047228408068431504, −3.39479211415431635584475863263, −1.05432585722005912438466835573, 3.82439471787428883458179771851, 4.68165747415547758308272050664, 6.78815580720238517465817564475, 7.43717482932174707395617527326, 9.335521857943243815007110119138, 10.31009524855261133835054731369, 11.41163512394869991389858154134, 12.80794159742815020298353629904, 14.16833210229999146103152893418, 15.05200565178231891580390689810

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