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Properties

Label 2-98-49.22-c1-0-0
Degree $2$
Conductor $98$
Sign $-0.655 - 0.755i$
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  + (−0.900 + 0.433i)2-s + (−0.614 + 2.69i)3-s + (0.623 − 0.781i)4-s + (0.0678 − 0.297i)5-s + (−0.614 − 2.69i)6-s + (−1.53 + 2.15i)7-s + (−0.222 + 0.974i)8-s + (−4.17 − 2.01i)9-s + (0.0678 + 0.297i)10-s + (−2.90 + 1.39i)11-s + (1.72 + 2.16i)12-s + (3.16 − 1.52i)13-s + (0.451 − 2.60i)14-s + (0.759 + 0.365i)15-s + (−0.222 − 0.974i)16-s + (2.75 + 3.45i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (−0.354 + 1.55i)3-s + (0.311 − 0.390i)4-s + (0.0303 − 0.132i)5-s + (−0.251 − 1.09i)6-s + (−0.581 + 0.813i)7-s + (−0.0786 + 0.344i)8-s + (−1.39 − 0.670i)9-s + (0.0214 + 0.0940i)10-s + (−0.875 + 0.421i)11-s + (0.497 + 0.623i)12-s + (0.877 − 0.422i)13-s + (0.120 − 0.696i)14-s + (0.196 + 0.0944i)15-s + (−0.0556 − 0.243i)16-s + (0.668 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.655 - 0.755i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ -0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259512 + 0.568929i\)
\(L(\frac12)\) \(\approx\) \(0.259512 + 0.568929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (1.53 - 2.15i)T \)
good3 \( 1 + (0.614 - 2.69i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-0.0678 + 0.297i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (2.90 - 1.39i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.16 + 1.52i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-2.75 - 3.45i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 - 7.35T + 19T^{2} \)
23 \( 1 + (-0.386 + 0.484i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (5.75 + 7.21i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 0.217T + 31T^{2} \)
37 \( 1 + (-3.50 - 4.39i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (1.74 - 7.66i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (1.05 + 4.61i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-6.67 + 3.21i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.591 + 0.741i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.27 + 5.56i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (5.07 + 6.37i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + (-4.03 + 5.05i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (7.99 + 3.85i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 6.04T + 79T^{2} \)
83 \( 1 + (-3.28 - 1.58i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-12.1 - 5.83i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 1.31T + 97T^{2} \)
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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

−14.99100173033303337412811943115, −13.28012801472510811815981279713, −11.87317188884400160753264664522, −10.75334012443114840144071130272, −9.894436341604319712982070625142, −9.179795576253223842732473552177, −7.923356017175900164421554789702, −5.98179635224338492923682437541, −5.14310061471526144701675562584, −3.30626689341149075080673669030, 1.02501078577428850261542856816, 3.08056250067361754221766295148, 5.75607211685300892568320299091, 7.11920773496376352199069970530, 7.60476455832937006700665357753, 9.084898049237820122636388680554, 10.52092322779339687703855615333, 11.44879945045619998142407724012, 12.48627000260541629276821192585, 13.38902565238735505940107396489

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