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Properties

Label 2-98-49.36-c1-0-3
Degree $2$
Conductor $98$
Sign $0.996 - 0.0822i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
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  + (−0.222 − 0.974i)2-s + (1.98 + 2.48i)3-s + (−0.900 + 0.433i)4-s + (−1.57 − 1.98i)5-s + (1.98 − 2.48i)6-s + (2.54 + 0.729i)7-s + (0.623 + 0.781i)8-s + (−1.58 + 6.93i)9-s + (−1.57 + 1.98i)10-s + (−0.406 − 1.78i)11-s + (−2.86 − 1.37i)12-s + (−0.451 − 1.97i)13-s + (0.145 − 2.64i)14-s + (1.79 − 7.85i)15-s + (0.623 − 0.781i)16-s + (−6.55 − 3.15i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (1.14 + 1.43i)3-s + (−0.450 + 0.216i)4-s + (−0.706 − 0.886i)5-s + (0.809 − 1.01i)6-s + (0.961 + 0.275i)7-s + (0.220 + 0.276i)8-s + (−0.527 + 2.31i)9-s + (−0.499 + 0.626i)10-s + (−0.122 − 0.536i)11-s + (−0.827 − 0.398i)12-s + (−0.125 − 0.548i)13-s + (0.0388 − 0.706i)14-s + (0.463 − 2.02i)15-s + (0.155 − 0.195i)16-s + (−1.58 − 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.996 - 0.0822i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.996 - 0.0822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15059 + 0.0473800i\)
\(L(\frac12)\) \(\approx\) \(1.15059 + 0.0473800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (-2.54 - 0.729i)T \)
good3 \( 1 + (-1.98 - 2.48i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (1.57 + 1.98i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (0.406 + 1.78i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.451 + 1.97i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (6.55 + 3.15i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
23 \( 1 + (-0.839 + 0.404i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-2.47 - 1.19i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 2.32T + 31T^{2} \)
37 \( 1 + (-3.19 - 1.53i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-4.24 - 5.32i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (4.86 - 6.09i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.469 - 2.05i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.11 + 1.50i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-1.07 + 1.34i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (7.74 + 3.72i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 - 8.17T + 67T^{2} \)
71 \( 1 + (-2.83 + 1.36i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (2.98 - 13.0i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 1.17T + 79T^{2} \)
83 \( 1 + (-2.31 + 10.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.28 - 10.0i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 - 0.616T + 97T^{2} \)
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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

−14.00730885608753404917001677867, −12.96244707847371633004791580863, −11.48283762626298586625148958891, −10.75603684163249097940001211510, −9.464779283991287508981360671631, −8.536397469689396642986111735773, −8.127159754840719924565302264808, −4.89383072705718762110256353493, −4.29468667379615094837232662188, −2.69887134038102981899322905485, 2.15533754452796691484369735323, 4.12354757883639383558896850307, 6.56942233866730383255319480772, 7.23825421749636968617244470849, 8.106179368419217033166832778041, 8.944178202161027380945863390713, 10.78445725415793381638655526690, 12.00712996337081709072265394460, 13.21263942737377030345679177729, 14.04062326835971369166108593027

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