↑

Properties

Label 2-70-35.18-c2-0-1
Degree $2$
Conductor $70$
Sign $0.687 + 0.725i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−5.05 + 1.35i)3-s + (1.73 + i)4-s + (4.84 − 1.23i)5-s + 7.39·6-s + (1.26 − 6.88i)7-s + (−1.99 − 2i)8-s + (15.8 − 9.16i)9-s + (−7.07 − 0.0936i)10-s + (5.56 − 9.64i)11-s + (−10.1 − 2.70i)12-s + (9.62 + 9.62i)13-s + (−4.25 + 8.94i)14-s + (−22.8 + 12.7i)15-s + (1.99 + 3.46i)16-s + (−2.29 − 8.55i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−1.68 + 0.451i)3-s + (0.433 + 0.250i)4-s + (0.969 − 0.246i)5-s + 1.23·6-s + (0.181 − 0.983i)7-s + (−0.249 − 0.250i)8-s + (1.76 − 1.01i)9-s + (−0.707 − 0.00936i)10-s + (0.506 − 0.876i)11-s + (−0.841 − 0.225i)12-s + (0.740 + 0.740i)13-s + (−0.303 + 0.638i)14-s + (−1.52 + 0.851i)15-s + (0.124 + 0.216i)16-s + (−0.134 − 0.503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.687 + 0.725i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.687 + 0.725i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.592232 - 0.254662i\)
\(L(\frac12)\) \(\approx\) \(0.592232 - 0.254662i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
5 \( 1 + (-4.84 + 1.23i)T \)
7 \( 1 + (-1.26 + 6.88i)T \)
good3 \( 1 + (5.05 - 1.35i)T + (7.79 - 4.5i)T^{2} \)
11 \( 1 + (-5.56 + 9.64i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-9.62 - 9.62i)T + 169iT^{2} \)
17 \( 1 + (2.29 + 8.55i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (5.79 - 3.34i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.37 + 27.5i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 29.0iT - 841T^{2} \)
31 \( 1 + (11.9 - 20.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (14.9 + 4.01i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 9.18T + 1.68e3T^{2} \)
43 \( 1 + (-55.1 - 55.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-8.55 - 2.29i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-13.8 + 3.69i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-67.7 - 39.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (10.7 + 18.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-13.7 - 51.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 101.T + 5.04e3T^{2} \)
73 \( 1 + (101. - 27.1i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (11.7 - 6.81i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-28.4 - 28.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (6.56 - 3.78i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-74.4 + 74.4i)T - 9.40e3iT^{2} \)
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

−14.23556457228354558843810188002, −13.01071172733829426726119933910, −11.65665442858456498714462306048, −10.87091066026654202247414355085, −10.10912447787739286602147395519, −8.878723893836721308077086107996, −6.80932824096324545775930108538, −5.94635534920245887400273621639, −4.35973074840989362381215496509, −0.982621961232194281795932307422, 1.66957993569840217454725678751, 5.35489780144568406195060000576, 6.08936180640168913673740925466, 7.19054181205584751045295252488, 9.013113693769120650060651668670, 10.28261093601380008383943834972, 11.15621467859496905041500501865, 12.22650033885301278133693879841, 13.15556948108208030269289117765, 14.90658029455703333613567735971

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