Properties

Label 2-98-49.15-c3-0-13
Degree $2$
Conductor $98$
Sign $-0.914 + 0.404i$
Analytic cond. $5.78218$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.445 − 1.94i)2-s + (6.02 − 7.54i)3-s + (−3.60 − 1.73i)4-s + (−10.1 + 12.6i)5-s + (−12.0 − 15.0i)6-s + (−9.34 − 15.9i)7-s + (−4.98 + 6.25i)8-s + (−14.7 − 64.5i)9-s + (20.2 + 25.3i)10-s + (11.9 − 52.3i)11-s + (−34.8 + 16.7i)12-s + (−8.74 + 38.3i)13-s + (−35.3 + 11.1i)14-s + (34.8 + 152. i)15-s + (9.97 + 12.5i)16-s + (41.9 − 20.1i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (1.15 − 1.45i)3-s + (−0.450 − 0.216i)4-s + (−0.903 + 1.13i)5-s + (−0.819 − 1.02i)6-s + (−0.504 − 0.863i)7-s + (−0.220 + 0.276i)8-s + (−0.546 − 2.39i)9-s + (0.639 + 0.801i)10-s + (0.327 − 1.43i)11-s + (−0.837 + 0.403i)12-s + (−0.186 + 0.817i)13-s + (−0.674 + 0.212i)14-s + (0.599 + 2.62i)15-s + (0.155 + 0.195i)16-s + (0.598 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.914 + 0.404i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ -0.914 + 0.404i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.369770 - 1.74870i\)
\(L(\frac12)\) \(\approx\) \(0.369770 - 1.74870i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.445 + 1.94i)T \)
7 \( 1 + (9.34 + 15.9i)T \)
good3 \( 1 + (-6.02 + 7.54i)T + (-6.00 - 26.3i)T^{2} \)
5 \( 1 + (10.1 - 12.6i)T + (-27.8 - 121. i)T^{2} \)
11 \( 1 + (-11.9 + 52.3i)T + (-1.19e3 - 577. i)T^{2} \)
13 \( 1 + (8.74 - 38.3i)T + (-1.97e3 - 953. i)T^{2} \)
17 \( 1 + (-41.9 + 20.1i)T + (3.06e3 - 3.84e3i)T^{2} \)
19 \( 1 - 79.0T + 6.85e3T^{2} \)
23 \( 1 + (-112. - 54.1i)T + (7.58e3 + 9.51e3i)T^{2} \)
29 \( 1 + (-83.9 + 40.4i)T + (1.52e4 - 1.90e4i)T^{2} \)
31 \( 1 - 142.T + 2.97e4T^{2} \)
37 \( 1 + (166. - 80.0i)T + (3.15e4 - 3.96e4i)T^{2} \)
41 \( 1 + (25.1 - 31.5i)T + (-1.53e4 - 6.71e4i)T^{2} \)
43 \( 1 + (-80.6 - 101. i)T + (-1.76e4 + 7.75e4i)T^{2} \)
47 \( 1 + (-29.7 + 130. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (359. + 173. i)T + (9.28e4 + 1.16e5i)T^{2} \)
59 \( 1 + (291. + 365. i)T + (-4.57e4 + 2.00e5i)T^{2} \)
61 \( 1 + (-371. + 179. i)T + (1.41e5 - 1.77e5i)T^{2} \)
67 \( 1 - 563.T + 3.00e5T^{2} \)
71 \( 1 + (174. + 83.8i)T + (2.23e5 + 2.79e5i)T^{2} \)
73 \( 1 + (-250. - 1.09e3i)T + (-3.50e5 + 1.68e5i)T^{2} \)
79 \( 1 + 1.32e3T + 4.93e5T^{2} \)
83 \( 1 + (25.0 + 109. i)T + (-5.15e5 + 2.48e5i)T^{2} \)
89 \( 1 + (-190. - 836. i)T + (-6.35e5 + 3.05e5i)T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−13.15005822479682845980940455078, −11.88789921341966493951490961178, −11.21773093115681664438422804130, −9.658319387161365834578924217896, −8.358866529672092161500720582895, −7.32343821955418938848027347432, −6.53845643846290647561410829477, −3.54128005665458356666450113151, −3.00487594828947717374677156038, −0.930330573635444030872544265477, 3.14447293946038509720692788207, 4.45387252324646412783106767326, 5.22797258877773938574474669447, 7.57043281868927692544546277291, 8.554884957217427953110632277100, 9.259911516700131866866192540687, 10.16128166787205776921712225795, 12.09206529000489556650446755782, 12.89174535540788008227896885498, 14.33089458832397473655222892954

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