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Properties

Label 2-70-7.2-c1-0-2
Degree $2$
Conductor $70$
Sign $0.386 + 0.922i$
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  + (−0.5 + 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 3·6-s + (0.5 − 2.59i)7-s + 0.999·8-s + (−3 + 5.19i)9-s + (0.499 + 0.866i)10-s + (1 + 1.73i)11-s + (−1.50 + 2.59i)12-s + (2 + 1.73i)14-s − 3·15-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + (−3 − 5.19i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 1.22·6-s + (0.188 − 0.981i)7-s + 0.353·8-s + (−1 + 1.73i)9-s + (0.158 + 0.273i)10-s + (0.301 + 0.522i)11-s + (−0.433 + 0.749i)12-s + (0.534 + 0.462i)14-s − 0.774·15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.707 − 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516716 - 0.343710i\)
\(L(\frac12)\) \(\approx\) \(0.516716 - 0.343710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−14.22179163347331973395479286369, −13.43321048577556451783639481960, −12.45153216538451057585401446535, −11.32160809582026924945694011020, −10.01876422695680704575392874056, −8.273069138088281899487541374925, −7.23818136825181266884753083364, −6.39523792820637389222306776151, −4.96869539482832325637312592115, −1.24616917281697553831129700746, 3.25116222547113681292297457491, 4.91249477605442021492006284165, 6.11722583319942204086745839998, 8.456590398572149363390766234868, 9.689065129551996658447007666514, 10.31547246634823912026371521725, 11.59553754720365939075670369519, 12.01490400259450215735294584831, 14.00886530662378715924612896919, 15.11319185182815227985695647685

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