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Properties

Label 2-98-7.4-c5-0-3
Degree $2$
Conductor $98$
Sign $-0.605 - 0.795i$
Analytic cond. $15.7176$
Root an. cond. $3.96454$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (−5.38 + 9.33i)3-s + (−7.99 + 13.8i)4-s + (53.0 + 91.8i)5-s + 43.1·6-s + 63.9·8-s + (63.4 + 109. i)9-s + (212. − 367. i)10-s + (46.7 − 80.9i)11-s + (−86.2 − 149. i)12-s − 661.·13-s − 1.14e3·15-s + (−128 − 221. i)16-s + (227. − 394. i)17-s + (253. − 439. i)18-s + (553. + 958. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.345 + 0.598i)3-s + (−0.249 + 0.433i)4-s + (0.949 + 1.64i)5-s + 0.488·6-s + 0.353·8-s + (0.261 + 0.452i)9-s + (0.671 − 1.16i)10-s + (0.116 − 0.201i)11-s + (−0.172 − 0.299i)12-s − 1.08·13-s − 1.31·15-s + (−0.125 − 0.216i)16-s + (0.191 − 0.331i)17-s + (0.184 − 0.319i)18-s + (0.351 + 0.608i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(15.7176\)
Root analytic conductor: \(3.96454\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :5/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.518522 + 1.04598i\)
\(L(\frac12)\) \(\approx\) \(0.518522 + 1.04598i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 3.46i)T \)
7 \( 1 \)
good3 \( 1 + (5.38 - 9.33i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-53.0 - 91.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-46.7 + 80.9i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 661.T + 3.71e5T^{2} \)
17 \( 1 + (-227. + 394. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-553. - 958. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (374. + 648. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.80e3T + 2.05e7T^{2} \)
31 \( 1 + (179. - 311. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.40e3 - 5.90e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 2.31e3T + 1.15e8T^{2} \)
43 \( 1 + 1.99e4T + 1.47e8T^{2} \)
47 \( 1 + (7.10e3 + 1.23e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.30e4 - 2.26e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.45e3 + 4.24e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (6.60e3 + 1.14e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.98e4 + 5.16e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 8.90e3T + 1.80e9T^{2} \)
73 \( 1 + (5.24e3 - 9.07e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.61e3 - 6.26e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.00e5T + 3.93e9T^{2} \)
89 \( 1 + (-1.00e4 - 1.73e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 2.33e4T + 8.58e9T^{2} \)
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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

−13.45194098355233281806216717915, −11.93030251435847493292077353129, −10.90878602379522242771705759281, −10.13159842594239430231853407087, −9.642584545405549780748851800518, −7.72888879986489296477465380872, −6.52661400295142244999929077192, −5.06251819222269856068996571252, −3.27887287788636323624960828640, −2.05813125337838751146138704357, 0.53633380921066211402810393333, 1.71099470701089151478495293609, 4.66849571635650954349337435761, 5.65673754240543201174100182744, 6.81576035987276479707771697091, 8.134091426564026703097449298705, 9.315312671297431920887052931016, 9.914519296389467440492520184751, 11.87474602161372918978736689411, 12.72055648969773570622861845883

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