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Properties

Label 2-70-35.17-c1-0-0
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: $\chi_{70} (17, \cdot )$
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

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

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