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Properties

Label 2-98-7.3-c6-0-5
Degree $2$
Conductor $98$
Sign $0.0633 - 0.997i$
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 + (−21.8 − 12.6i)3-s + (−15.9 + 27.7i)4-s + (10.7 − 6.20i)5-s − 142. i·6-s − 181.·8-s + (−45.0 − 78.0i)9-s + (60.8 + 35.1i)10-s + (774. − 1.34e3i)11-s + (700. − 404. i)12-s + 2.77e3i·13-s − 313.·15-s + (−512. − 886. i)16-s + (109. + 63.2i)17-s + (254. − 441. i)18-s + (1.14e3 − 662. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.810 − 0.468i)3-s + (−0.249 + 0.433i)4-s + (0.0859 − 0.0496i)5-s − 0.661i·6-s − 0.353·8-s + (−0.0618 − 0.107i)9-s + (0.0608 + 0.0351i)10-s + (0.582 − 1.00i)11-s + (0.405 − 0.234i)12-s + 1.26i·13-s − 0.0929·15-s + (−0.125 − 0.216i)16-s + (0.0223 + 0.0128i)17-s + (0.0437 − 0.0757i)18-s + (0.167 − 0.0966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(22.5453\)
Root analytic conductor: \(4.74818\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.02036 + 0.957662i\)
\(L(\frac12)\) \(\approx\) \(1.02036 + 0.957662i\)
\(L(4)\) 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 \( 1 + (21.8 + 12.6i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-10.7 + 6.20i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-774. + 1.34e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 2.77e3iT - 4.82e6T^{2} \)
17 \( 1 + (-109. - 63.2i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-1.14e3 + 662. i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-7.11e3 - 1.23e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 7.47e3T + 5.94e8T^{2} \)
31 \( 1 + (-4.92e4 - 2.84e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-4.50e4 - 7.79e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 3.57e4iT - 4.75e9T^{2} \)
43 \( 1 + 7.94e4T + 6.32e9T^{2} \)
47 \( 1 + (1.26e5 - 7.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-8.50e4 + 1.47e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.87e5 - 1.08e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.72e5 + 9.95e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (7.22e4 - 1.25e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 4.07e5T + 1.28e11T^{2} \)
73 \( 1 + (-1.61e5 - 9.33e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (4.19e4 + 7.26e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 1.62e5iT - 3.26e11T^{2} \)
89 \( 1 + (-4.39e5 + 2.53e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 5.09e5iT - 8.32e11T^{2} \)
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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

−13.14528545717628033934679119146, −11.71554149933840229709661540940, −11.46588922885292297725802859069, −9.564143584157186456261647362877, −8.458750791338761867921671238528, −6.93306933466952358973943063137, −6.23669036496236809832677886788, −5.07279129683708988832299711552, −3.47444248043159321998716965194, −1.16910421059636347317068190850, 0.56319121329904056887184076538, 2.45952551185770016500390009496, 4.20015302226040503395212860351, 5.22879368683612283053134107360, 6.39324924095235911560710468750, 8.092090473723925838490992161148, 9.732290081385541852377130671224, 10.41159201540023193020944778966, 11.44007706042292646930001859860, 12.31843741583166925688174874285

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