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Properties

Label 2-98-7.2-c3-0-6
Degree $2$
Conductor $98$
Sign $0.827 + 0.561i$
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  + (1 − 1.73i)2-s + (3.53 + 6.12i)3-s + (−1.99 − 3.46i)4-s + (9.89 − 17.1i)5-s + 14.1·6-s − 7.99·8-s + (−11.5 + 19.9i)9-s + (−19.7 − 34.2i)10-s + (7 + 12.1i)11-s + (14.1 − 24.4i)12-s + 50.9·13-s + 140·15-s + (−8 + 13.8i)16-s + (−0.707 − 1.22i)17-s + (22.9 + 39.8i)18-s + (0.707 − 1.22i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.680 + 1.17i)3-s + (−0.249 − 0.433i)4-s + (0.885 − 1.53i)5-s + 0.962·6-s − 0.353·8-s + (−0.425 + 0.737i)9-s + (−0.626 − 1.08i)10-s + (0.191 + 0.332i)11-s + (0.340 − 0.589i)12-s + 1.08·13-s + 2.40·15-s + (−0.125 + 0.216i)16-s + (−0.0100 − 0.0174i)17-s + (0.301 + 0.521i)18-s + (0.00853 − 0.0147i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.39971 - 0.737359i\)
\(L(\frac12)\) \(\approx\) \(2.39971 - 0.737359i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
7 \( 1 \)
good3 \( 1 + (-3.53 - 6.12i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-9.89 + 17.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7 - 12.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 50.9T + 2.19e3T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-0.707 + 1.22i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (70 - 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 286T + 2.43e4T^{2} \)
31 \( 1 + (-46.6 - 80.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-19 + 32.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 125.T + 6.89e4T^{2} \)
43 \( 1 + 34T + 7.95e4T^{2} \)
47 \( 1 + (261. - 453. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-37 - 64.0i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (217. + 375. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (7.07 - 12.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (342 + 592. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 588T + 3.57e5T^{2} \)
73 \( 1 + (-135. - 233. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (610 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 422.T + 5.71e5T^{2} \)
89 \( 1 + (309. - 535. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−13.37955571283272456360069313875, −12.47088979928303003074391051223, −11.09197514549841196593492503454, −9.752198057551338075762201187225, −9.339528961436438735939793019691, −8.362076369799738357289211538753, −5.81452154018555564508351637710, −4.73046188495361210215353289188, −3.64676334064836411995569995788, −1.62109349757962770749185104657, 2.08253272346669541163945260014, 3.42490691313573738738284499624, 5.97000276392372530434002676072, 6.65691682630307021780731236259, 7.64110441500088767462483831442, 8.811104789708765217865738701935, 10.30850857420329854535453550365, 11.53554961328440938915046031909, 13.08259374526230461876347408971, 13.61426101605523301797944118469

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