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Properties

Label 2-98-49.4-c1-0-2
Degree $2$
Conductor $98$
Sign $0.801 - 0.598i$
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  + (0.826 + 0.563i)2-s + (0.0584 + 0.0541i)3-s + (0.365 + 0.930i)4-s + (−0.427 + 0.131i)5-s + (0.0177 + 0.0776i)6-s + (1.60 + 2.10i)7-s + (−0.222 + 0.974i)8-s + (−0.223 − 2.98i)9-s + (−0.427 − 0.131i)10-s + (0.349 − 4.66i)11-s + (−0.0291 + 0.0741i)12-s + (−6.27 + 3.02i)13-s + (0.141 + 2.64i)14-s + (−0.0320 − 0.0154i)15-s + (−0.733 + 0.680i)16-s + (1.31 − 0.198i)17-s + ⋯
L(s)  = 1  + (0.584 + 0.398i)2-s + (0.0337 + 0.0312i)3-s + (0.182 + 0.465i)4-s + (−0.190 + 0.0589i)5-s + (0.00723 + 0.0317i)6-s + (0.606 + 0.794i)7-s + (−0.0786 + 0.344i)8-s + (−0.0745 − 0.995i)9-s + (−0.135 − 0.0416i)10-s + (0.105 − 1.40i)11-s + (−0.00840 + 0.0214i)12-s + (−1.73 + 0.837i)13-s + (0.0377 + 0.706i)14-s + (−0.00828 − 0.00398i)15-s + (−0.183 + 0.170i)16-s + (0.319 − 0.0481i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24250 + 0.412561i\)
\(L(\frac12)\) \(\approx\) \(1.24250 + 0.412561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 - 0.563i)T \)
7 \( 1 + (-1.60 - 2.10i)T \)
good3 \( 1 + (-0.0584 - 0.0541i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (0.427 - 0.131i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.349 + 4.66i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (6.27 - 3.02i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-1.31 + 0.198i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (2.50 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.25 - 0.490i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-3.44 - 4.31i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-2.08 + 3.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.654 + 1.66i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (1.12 - 4.93i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-1.75 - 7.69i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-4.41 - 3.00i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (3.28 + 8.37i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-9.60 - 2.96i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (2.27 - 5.78i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (0.329 - 0.570i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.158 + 0.198i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (3.38 - 2.30i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (0.928 + 1.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.1 + 6.79i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.616 + 8.23i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−14.36693707179857427448793752962, −13.00508878561779921427819819058, −11.87443460000569792292346900054, −11.34104289318661664846145253003, −9.446496358490143978784289303622, −8.499967398055977668061288535284, −7.11492852310506890717872957945, −5.89049991844944013890541279354, −4.60388855221070353945580020910, −2.89747191992534239389739920089, 2.23227674732322735060424160587, 4.31433493259252096327581320113, 5.20022497772438251238229363939, 7.17323602719712672790818999529, 7.989659659793693091217958652224, 10.02652671499762397086953656448, 10.47825454293748260676935205742, 11.96808870372146671716218852958, 12.63125384783796654315977889076, 13.87517603118903386691938079891

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