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Properties

Label 2-98-49.36-c1-0-5
Degree $2$
Conductor $98$
Sign $-0.577 + 0.816i$
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 + (−0.683 − 0.856i)3-s + (−0.900 + 0.433i)4-s + (−1.25 − 1.57i)5-s + (−0.683 + 0.856i)6-s + (−0.413 − 2.61i)7-s + (0.623 + 0.781i)8-s + (0.400 − 1.75i)9-s + (−1.25 + 1.57i)10-s + (1.37 + 6.04i)11-s + (0.986 + 0.475i)12-s + (−0.794 − 3.48i)13-s + (−2.45 + 0.984i)14-s + (−0.489 + 2.14i)15-s + (0.623 − 0.781i)16-s + (1.98 + 0.956i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (−0.394 − 0.494i)3-s + (−0.450 + 0.216i)4-s + (−0.560 − 0.702i)5-s + (−0.278 + 0.349i)6-s + (−0.156 − 0.987i)7-s + (0.220 + 0.276i)8-s + (0.133 − 0.584i)9-s + (−0.396 + 0.496i)10-s + (0.416 + 1.82i)11-s + (0.284 + 0.137i)12-s + (−0.220 − 0.965i)13-s + (−0.656 + 0.263i)14-s + (−0.126 + 0.554i)15-s + (0.155 − 0.195i)16-s + (0.481 + 0.232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.577 + 0.816i$
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: \((2,\ 98,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333999 - 0.645469i\)
\(L(\frac12)\) \(\approx\) \(0.333999 - 0.645469i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.413 + 2.61i)T \)
good3 \( 1 + (0.683 + 0.856i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (1.25 + 1.57i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (-1.37 - 6.04i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.794 + 3.48i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.98 - 0.956i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 8.01T + 19T^{2} \)
23 \( 1 + (1.24 - 0.600i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-4.97 - 2.39i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 + (-1.26 - 0.611i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (3.94 + 4.95i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (4.02 - 5.04i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (0.417 + 1.82i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-4.33 + 2.08i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (2.43 - 3.05i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (0.977 + 0.470i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (6.72 - 3.23i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.576 - 2.52i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (1.21 - 5.32i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.13 - 9.36i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−13.11007186185827393679050793245, −12.32389440921088664979446578079, −11.82192252102199910358988669091, −10.25184171313941341657281308047, −9.513262016148176524516325247199, −7.86179697492133896970515993372, −7.00024371627743045354866342717, −5.01018294521160739972069337407, −3.69447799902218817388611534207, −1.08671803123315267187964156584, 3.37072127918611308395112629026, 5.15964092723759428137759643139, 6.21066404236417097519204507275, 7.57330220758901150261694924999, 8.764300634909516794450733717571, 9.877527927283264396723328850536, 11.26229071521031660037602380830, 11.80319518260394906884668266763, 13.64948475834919206410584431550, 14.36095936551650146005539758419

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