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Properties

Label 2-98-7.6-c4-0-8
Degree $2$
Conductor $98$
Sign $0.912 + 0.409i$
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  + 2.82·2-s − 2.03i·3-s + 8.00·4-s − 0.710i·5-s − 5.75i·6-s + 22.6·8-s + 76.8·9-s − 2.01i·10-s + 151.·11-s − 16.2i·12-s − 260. i·13-s − 1.44·15-s + 64.0·16-s + 385. i·17-s + 217.·18-s − 390. i·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.225i·3-s + 0.500·4-s − 0.0284i·5-s − 0.159i·6-s + 0.353·8-s + 0.948·9-s − 0.0201i·10-s + 1.25·11-s − 0.112i·12-s − 1.54i·13-s − 0.00642·15-s + 0.250·16-s + 1.33i·17-s + 0.671·18-s − 1.08i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(10.1302\)
Root analytic conductor: \(3.18280\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :2),\ 0.912 + 0.409i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.81841 - 0.602895i\)
\(L(\frac12)\) \(\approx\) \(2.81841 - 0.602895i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
7 \( 1 \)
good3 \( 1 + 2.03iT - 81T^{2} \)
5 \( 1 + 0.710iT - 625T^{2} \)
11 \( 1 - 151.T + 1.46e4T^{2} \)
13 \( 1 + 260. iT - 2.85e4T^{2} \)
17 \( 1 - 385. iT - 8.35e4T^{2} \)
19 \( 1 + 390. iT - 1.30e5T^{2} \)
23 \( 1 + 177.T + 2.79e5T^{2} \)
29 \( 1 + 320.T + 7.07e5T^{2} \)
31 \( 1 - 1.34e3iT - 9.23e5T^{2} \)
37 \( 1 + 797.T + 1.87e6T^{2} \)
41 \( 1 + 815. iT - 2.82e6T^{2} \)
43 \( 1 + 2.16e3T + 3.41e6T^{2} \)
47 \( 1 - 4.28e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.17e3T + 7.89e6T^{2} \)
59 \( 1 + 4.70e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.53e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.09e3T + 2.01e7T^{2} \)
71 \( 1 + 2.25e3T + 2.54e7T^{2} \)
73 \( 1 - 4.65e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.19e3T + 3.89e7T^{2} \)
83 \( 1 - 7.79e3iT - 4.74e7T^{2} \)
89 \( 1 + 9.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.94e3iT - 8.85e7T^{2} \)
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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

−12.87277942496599435154858815129, −12.50350179670443281030142782496, −11.08461986547497709768286146518, −10.10246030504638013412553783309, −8.598581131374087411535289319703, −7.22440140434609308480334313288, −6.19708439058273027815872030910, −4.71415736044656992790782169838, −3.35609601312238948972969597268, −1.38005779904334613316057255077, 1.69652706475250238683626984229, 3.74233427024350712239302019594, 4.72101953687322744767948293647, 6.39914027838077466608269808699, 7.28524624367948847931163293005, 9.069151759505528984305007828844, 10.01928694434016692061239100244, 11.49732943880406797089984784521, 12.09259414254164852360098678194, 13.41032267394193362855505401846

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