Properties

Label 2-70-35.33-c1-0-3
Degree $2$
Conductor $70$
Sign $-0.963 + 0.269i$
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  + (−0.258 − 0.965i)2-s + (−2.80 − 0.752i)3-s + (−0.866 + 0.499i)4-s + (−2.21 − 0.318i)5-s + 2.90i·6-s + (0.559 − 2.58i)7-s + (0.707 + 0.707i)8-s + (4.71 + 2.72i)9-s + (0.264 + 2.22i)10-s + (−1.83 − 3.17i)11-s + (2.80 − 0.752i)12-s + (0.830 − 0.830i)13-s + (−2.64 + 0.128i)14-s + (5.97 + 2.55i)15-s + (0.500 − 0.866i)16-s + (−0.204 + 0.761i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−1.62 − 0.434i)3-s + (−0.433 + 0.249i)4-s + (−0.989 − 0.142i)5-s + 1.18i·6-s + (0.211 − 0.977i)7-s + (0.249 + 0.249i)8-s + (1.57 + 0.908i)9-s + (0.0837 + 0.702i)10-s + (−0.553 − 0.958i)11-s + (0.810 − 0.217i)12-s + (0.230 − 0.230i)13-s + (−0.706 + 0.0343i)14-s + (1.54 + 0.660i)15-s + (0.125 − 0.216i)16-s + (−0.0494 + 0.184i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.963 + 0.269i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ -0.963 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0456572 - 0.333129i\)
\(L(\frac12)\) \(\approx\) \(0.0456572 - 0.333129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (2.21 + 0.318i)T \)
7 \( 1 + (-0.559 + 2.58i)T \)
good3 \( 1 + (2.80 + 0.752i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.83 + 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.830 + 0.830i)T - 13iT^{2} \)
17 \( 1 + (0.204 - 0.761i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.09 + 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.54 - 1.21i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (-0.0359 + 0.0207i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0664 - 0.248i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 + 0.474i)T + 43iT^{2} \)
47 \( 1 + (-6.18 + 1.65i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.04 + 7.64i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.35 - 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 0.996i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.39 - 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (-9.52 - 2.55i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.6 + 6.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.73 - 9.73i)T - 83iT^{2} \)
89 \( 1 + (-0.715 + 1.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−13.77302305888301279424868435283, −12.78425065079318357404386140655, −11.73529006881836466953832527948, −11.08853542864648273027524357028, −10.31636041995121553237801115212, −8.229815270820594082152431361808, −7.09399189403550492169163516773, −5.45451210884874825834839254380, −4.00044006127484533342157648903, −0.60306672000232762986585915702, 4.43541299963985890664564840216, 5.49412612352023004222887606811, 6.73251047729020077892503605280, 8.069548945135435342703124880071, 9.684069750730351042695130676955, 10.87617426620490241195706382810, 11.89015636033850439148501915017, 12.60067230886931552114249324559, 14.63144393740684633682996015002, 15.67556920818155896936872292396

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