↑

Properties

Label 2-70-5.4-c5-0-9
Degree $2$
Conductor $70$
Sign $0.776 + 0.630i$
Analytic cond. $11.2268$
Root an. cond. $3.35065$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 22.6i·3-s − 16·4-s + (43.3 + 35.2i)5-s + 90.7·6-s + 49i·7-s − 64i·8-s − 272.·9-s + (−141. + 173. i)10-s + 527.·11-s + 363. i·12-s − 1.08e3i·13-s − 196·14-s + (800. − 984. i)15-s + 256·16-s − 1.82e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.45i·3-s − 0.5·4-s + (0.776 + 0.630i)5-s + 1.02·6-s + 0.377i·7-s − 0.353i·8-s − 1.11·9-s + (−0.445 + 0.548i)10-s + 1.31·11-s + 0.727i·12-s − 1.77i·13-s − 0.267·14-s + (0.918 − 1.12i)15-s + 0.250·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(11.2268\)
Root analytic conductor: \(3.35065\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :5/2),\ 0.776 + 0.630i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.80762 - 0.641812i\)
\(L(\frac12)\) \(\approx\) \(1.80762 - 0.641812i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (-43.3 - 35.2i)T \)
7 \( 1 - 49iT \)
good3 \( 1 + 22.6iT - 243T^{2} \)
11 \( 1 - 527.T + 1.61e5T^{2} \)
13 \( 1 + 1.08e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.82e3iT - 1.41e6T^{2} \)
19 \( 1 - 113.T + 2.47e6T^{2} \)
23 \( 1 - 8.45iT - 6.43e6T^{2} \)
29 \( 1 - 7.75e3T + 2.05e7T^{2} \)
31 \( 1 + 5.07e3T + 2.86e7T^{2} \)
37 \( 1 - 1.12e4iT - 6.93e7T^{2} \)
41 \( 1 - 4.49e3T + 1.15e8T^{2} \)
43 \( 1 + 1.71e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.65e3iT - 2.29e8T^{2} \)
53 \( 1 - 6.20e3iT - 4.18e8T^{2} \)
59 \( 1 + 6.32e3T + 7.14e8T^{2} \)
61 \( 1 - 2.28e4T + 8.44e8T^{2} \)
67 \( 1 - 6.60e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.98e4T + 1.80e9T^{2} \)
73 \( 1 - 2.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.27e4T + 3.07e9T^{2} \)
83 \( 1 - 8.15e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.01e4T + 5.58e9T^{2} \)
97 \( 1 + 9.37e4iT - 8.58e9T^{2} \)
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

−13.72456588792517806966711737774, −12.74679281313366293520625524818, −11.69313174057816816675008232791, −10.01293081228602045489840377939, −8.662666024089320744573184540762, −7.35497334903504880372293437000, −6.55084057257130076573251874012, −5.49575929342288443442393764266, −2.80566782478734642695123587448, −0.992084140478228599584514691318, 1.61149857869042726655174136103, 3.88243140363068982158760594627, 4.60372037360518008425324630800, 6.27020317734126453716070164450, 8.809272923373119837507441143511, 9.390976237854387166037473728439, 10.33452324337145829531893188336, 11.39031198771377445161699193733, 12.59817541572690969187889117463, 14.04578028264643251466809573307

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