Properties

Label 2-98-49.32-c1-0-0
Degree $2$
Conductor $98$
Sign $-0.986 - 0.160i$
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  + (0.0747 + 0.997i)2-s + (−2.39 − 0.739i)3-s + (−0.988 + 0.149i)4-s + (−2.18 + 2.02i)5-s + (0.557 − 2.44i)6-s + (−2.12 + 1.56i)7-s + (−0.222 − 0.974i)8-s + (2.71 + 1.85i)9-s + (−2.18 − 2.02i)10-s + (1.16 − 0.791i)11-s + (2.47 + 0.373i)12-s + (0.0398 + 0.0191i)13-s + (−1.72 − 2.00i)14-s + (6.73 − 3.24i)15-s + (0.955 − 0.294i)16-s + (2.61 + 6.66i)17-s + ⋯
L(s)  = 1  + (0.0528 + 0.705i)2-s + (−1.38 − 0.426i)3-s + (−0.494 + 0.0745i)4-s + (−0.977 + 0.907i)5-s + (0.227 − 0.997i)6-s + (−0.804 + 0.593i)7-s + (−0.0786 − 0.344i)8-s + (0.905 + 0.617i)9-s + (−0.691 − 0.641i)10-s + (0.350 − 0.238i)11-s + (0.715 + 0.107i)12-s + (0.0110 + 0.00532i)13-s + (−0.460 − 0.536i)14-s + (1.73 − 0.837i)15-s + (0.238 − 0.0736i)16-s + (0.634 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.986 - 0.160i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ -0.986 - 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0237005 + 0.292886i\)
\(L(\frac12)\) \(\approx\) \(0.0237005 + 0.292886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (2.12 - 1.56i)T \)
good3 \( 1 + (2.39 + 0.739i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (2.18 - 2.02i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (-1.16 + 0.791i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (-0.0398 - 0.0191i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-2.61 - 6.66i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (3.92 + 6.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 - 5.40i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (2.49 - 3.12i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.20 - 3.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.811 + 0.122i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (0.416 + 1.82i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.802 + 3.51i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.368 + 4.92i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (5.60 - 0.845i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-10.2 - 9.49i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-2.31 - 0.349i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.01 - 2.52i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.429 + 5.72i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-4.13 - 7.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.263 - 0.126i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-8.78 - 5.98i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−14.77919500693287435034090940182, −13.15995316918039539151241197742, −12.27782863863720826679003397297, −11.36930905881284768614427335730, −10.42943353754628713429229979616, −8.750929896281030322184394097147, −7.24872214284244284259231778443, −6.50110652277057194184976353645, −5.54128900929376947744142169372, −3.68319470698248173944233023636, 0.39894934872393933643593385911, 3.88833236722869608703937342544, 4.80315820469413254647114540367, 6.23211170795296994177054153470, 7.88310000962094164244934976211, 9.482538927701501404772604515121, 10.37065162018421865509420352345, 11.48616609728689753845909258501, 12.18869820931545878745967216154, 12.85353468533956678690244587006

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