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Properties

Label 2-98-7.3-c4-0-5
Degree $2$
Conductor $98$
Sign $0.995 - 0.0956i$
Analytic cond. $10.1302$
Root an. cond. $3.18280$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 2.44i)2-s + (−13.8 − 7.99i)3-s + (−3.99 + 6.92i)4-s + (−24.1 + 13.9i)5-s − 45.2i·6-s − 22.6·8-s + (87.4 + 151. i)9-s + (−68.2 − 39.3i)10-s + (69.8 − 120. i)11-s + (110. − 63.9i)12-s − 101. i·13-s + 445.·15-s + (−32.0 − 55.4i)16-s + (470. + 271. i)17-s + (−247. + 428. i)18-s + (121. − 69.9i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.53 − 0.888i)3-s + (−0.249 + 0.433i)4-s + (−0.964 + 0.556i)5-s − 1.25i·6-s − 0.353·8-s + (1.07 + 1.86i)9-s + (−0.682 − 0.393i)10-s + (0.577 − 0.999i)11-s + (0.769 − 0.444i)12-s − 0.603i·13-s + 1.97·15-s + (−0.125 − 0.216i)16-s + (1.62 + 0.939i)17-s + (−0.763 + 1.32i)18-s + (0.335 − 0.193i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 - 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.995 - 0.0956i$
Analytic conductor: \(10.1302\)
Root analytic conductor: \(3.18280\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :2),\ 0.995 - 0.0956i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.920842 + 0.0441518i\)
\(L(\frac12)\) \(\approx\) \(0.920842 + 0.0441518i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 2.44i)T \)
7 \( 1 \)
good3 \( 1 + (13.8 + 7.99i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (24.1 - 13.9i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-69.8 + 120. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 101. iT - 2.85e4T^{2} \)
17 \( 1 + (-470. - 271. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-121. + 69.9i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (114. + 198. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 383.T + 7.07e5T^{2} \)
31 \( 1 + (-344. - 198. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-449. - 778. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 657. iT - 2.82e6T^{2} \)
43 \( 1 - 1.15e3T + 3.41e6T^{2} \)
47 \( 1 + (-1.12e3 + 647. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.58e3 + 4.47e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-3.59e3 - 2.07e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.59e3 - 920. i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.15e3 + 5.46e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.26e3T + 2.54e7T^{2} \)
73 \( 1 + (3.46e3 + 1.99e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-5.30e3 - 9.19e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 5.75e3iT - 4.74e7T^{2} \)
89 \( 1 + (3.62e3 - 2.09e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 3.81e3iT - 8.85e7T^{2} \)
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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

−13.01129463928488097782404192907, −12.08388968786262782286255606650, −11.44551096020442676422398700417, −10.40852610489606965071138126528, −8.174316538380106348201683457979, −7.33084435517907800599412216102, −6.25411301457732166092916726118, −5.40629114821830149293987972896, −3.64181725839252094400979193752, −0.73846319848340782076681592108, 0.866370246941730005744208344395, 3.87260877611879701440506984696, 4.67402167341596345931333643186, 5.75513523496487960605285210048, 7.35241942526691188595050560494, 9.357151676825618415304240909708, 10.10222128216211070353061834067, 11.41456957812643898392432263885, 11.92707987443881025340350504737, 12.50947240710252536022425841765

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