↑

Properties

Label 2-98-7.4-c3-0-9
Degree $2$
Conductor $98$
Sign $-0.991 + 0.126i$
Analytic cond. $5.78218$
Root an. cond. $2.40461$
Motivic weight $3$
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 − 1.73i)2-s + (4.69 − 8.12i)3-s + (−1.99 + 3.46i)4-s + (−4.69 − 8.12i)5-s − 18.7·6-s + 7.99·8-s + (−30.5 − 52.8i)9-s + (−9.38 + 16.2i)10-s + (−10 + 17.3i)11-s + (18.7 + 32.4i)12-s + 65.6·13-s − 88.0·15-s + (−8 − 13.8i)16-s + (−28.1 + 48.7i)17-s + (−61 + 105. i)18-s + (−4.69 − 8.12i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.902 − 1.56i)3-s + (−0.249 + 0.433i)4-s + (−0.419 − 0.726i)5-s − 1.27·6-s + 0.353·8-s + (−1.12 − 1.95i)9-s + (−0.296 + 0.513i)10-s + (−0.274 + 0.474i)11-s + (0.451 + 0.781i)12-s + 1.40·13-s − 1.51·15-s + (−0.125 − 0.216i)16-s + (−0.401 + 0.695i)17-s + (−0.798 + 1.38i)18-s + (−0.0566 − 0.0980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0915558 - 1.44272i\)
\(L(\frac12)\) \(\approx\) \(0.0915558 - 1.44272i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (-4.69 + 8.12i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (4.69 + 8.12i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 65.6T + 2.19e3T^{2} \)
17 \( 1 + (28.1 - 48.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (4.69 + 8.12i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (24 + 41.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 166T + 2.43e4T^{2} \)
31 \( 1 + (-103. + 178. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-39 - 67.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 393.T + 6.89e4T^{2} \)
43 \( 1 - 436T + 7.95e4T^{2} \)
47 \( 1 + (103. + 178. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (31 - 53.6i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-333. + 576. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (136. + 235. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (290 - 502. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 544T + 3.57e5T^{2} \)
73 \( 1 + (-300. + 519. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-340 - 588. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 196.T + 5.71e5T^{2} \)
89 \( 1 + (-750. - 1.29e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 656.T + 9.12e5T^{2} \)
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

−12.92623340602245963381738890110, −12.17367727827051415933211004844, −11.01736482175531653562721628363, −9.253498817837874511772572686556, −8.386520812354337881093607107443, −7.70781109034931983143823401402, −6.27285004242027547278664608027, −3.94049487770476489304774672973, −2.25659947204738864328203188092, −0.893627361577646493719601242722, 3.08169621306801976619009874574, 4.23185301754449447060188513828, 5.77558088722491183261578313606, 7.50743599629853213524003382053, 8.628291730027317161880456142578, 9.386772777887015292392450120436, 10.65732476846797292052499693117, 11.15037834653918480614456127576, 13.46789709666785119542865065822, 14.28591323521111707633396623002

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