↑

Properties

Label 2-70-7.4-c1-0-0
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

Related objects

Downloads

Learn more

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: $\chi_{70} (11, \cdot )$
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 \( 1 - 14T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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