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Properties

Label 2-98-7.5-c6-0-4
Degree $2$
Conductor $98$
Sign $0.832 - 0.553i$
Analytic cond. $22.5453$
Root an. cond. $4.74818$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.82 − 4.89i)2-s + (−3.32 + 1.92i)3-s + (−15.9 − 27.7i)4-s + (−177. − 102. i)5-s + 21.7i·6-s − 181.·8-s + (−357. + 618. i)9-s + (−1.00e3 + 579. i)10-s + (588. + 1.02e3i)11-s + (106. + 61.4i)12-s − 1.39e3i·13-s + 786.·15-s + (−512. + 886. i)16-s + (5.25e3 − 3.03e3i)17-s + (2.02e3 + 3.49e3i)18-s + (2.71e3 + 1.56e3i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.123 + 0.0711i)3-s + (−0.249 − 0.433i)4-s + (−1.41 − 0.818i)5-s + 0.100i·6-s − 0.353·8-s + (−0.489 + 0.848i)9-s + (−1.00 + 0.579i)10-s + (0.442 + 0.766i)11-s + (0.0616 + 0.0355i)12-s − 0.637i·13-s + 0.233·15-s + (−0.125 + 0.216i)16-s + (1.06 − 0.617i)17-s + (0.346 + 0.599i)18-s + (0.395 + 0.228i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(22.5453\)
Root analytic conductor: \(4.74818\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.951551 + 0.287572i\)
\(L(\frac12)\) \(\approx\) \(0.951551 + 0.287572i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 + 4.89i)T \)
7 \( 1 \)
good3 \( 1 + (3.32 - 1.92i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (177. + 102. i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-588. - 1.02e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 1.39e3iT - 4.82e6T^{2} \)
17 \( 1 + (-5.25e3 + 3.03e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-2.71e3 - 1.56e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (4.32e3 - 7.49e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 3.25e4T + 5.94e8T^{2} \)
31 \( 1 + (3.66e4 - 2.11e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (1.87e4 - 3.25e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 9.35e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.20e4T + 6.32e9T^{2} \)
47 \( 1 + (2.16e4 + 1.24e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (54.8 + 94.9i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (2.82e5 - 1.63e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-7.32e4 - 4.22e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-5.55e4 - 9.61e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 4.41e5T + 1.28e11T^{2} \)
73 \( 1 + (-4.02e5 + 2.32e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (2.97e5 - 5.15e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 6.54e5iT - 3.26e11T^{2} \)
89 \( 1 + (-2.58e5 - 1.49e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.69e5iT - 8.32e11T^{2} \)
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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

−12.43629917779569029572627910771, −11.98111586180032306904132216278, −10.97572931953264772429711537801, −9.726216790244836797504276560730, −8.365346873322550392977199804410, −7.46477183449705813820744108988, −5.39665519262693527558084934314, −4.46963492571098147940397054497, −3.16126089972675296988791530120, −1.14896801055440444561610513050, 0.37196464067113717996963616144, 3.23326300118375306686955584857, 4.05139789315218854089412243286, 5.91027665881994432767336802186, 6.92326590524762630553711640164, 7.960615906010964239063082204220, 9.044563041076082640065282474898, 10.79004586392223278007303076064, 11.77676217949884988708420956204, 12.39559612123998653065122018340

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