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Properties

Label 2-98-7.6-c6-0-11
Degree $2$
Conductor $98$
Sign $0.755 + 0.654i$
Analytic cond. $22.5453$
Root an. cond. $4.74818$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65·2-s − 31.8i·3-s + 32.0·4-s + 129. i·5-s − 180. i·6-s + 181.·8-s − 287.·9-s + 730. i·10-s + 2.37e3·11-s − 1.02e3i·12-s − 820. i·13-s + 4.11e3·15-s + 1.02e3·16-s + 2.63e3i·17-s − 1.62e3·18-s − 5.55e3i·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.18i·3-s + 0.500·4-s + 1.03i·5-s − 0.834i·6-s + 0.353·8-s − 0.394·9-s + 0.730i·10-s + 1.78·11-s − 0.590i·12-s − 0.373i·13-s + 1.21·15-s + 0.250·16-s + 0.536i·17-s − 0.278·18-s − 0.810i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(22.5453\)
Root analytic conductor: \(4.74818\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.18094 - 1.18593i\)
\(L(\frac12)\) \(\approx\) \(3.18094 - 1.18593i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65T \)
7 \( 1 \)
good3 \( 1 + 31.8iT - 729T^{2} \)
5 \( 1 - 129. iT - 1.56e4T^{2} \)
11 \( 1 - 2.37e3T + 1.77e6T^{2} \)
13 \( 1 + 820. iT - 4.82e6T^{2} \)
17 \( 1 - 2.63e3iT - 2.41e7T^{2} \)
19 \( 1 + 5.55e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.13e4T + 1.48e8T^{2} \)
29 \( 1 + 3.28e4T + 5.94e8T^{2} \)
31 \( 1 + 4.69e4iT - 8.87e8T^{2} \)
37 \( 1 - 2.17e4T + 2.56e9T^{2} \)
41 \( 1 - 8.58e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.13e5T + 6.32e9T^{2} \)
47 \( 1 - 3.85e4iT - 1.07e10T^{2} \)
53 \( 1 - 3.32e4T + 2.21e10T^{2} \)
59 \( 1 - 3.34e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.68e5iT - 5.15e10T^{2} \)
67 \( 1 + 3.22e5T + 9.04e10T^{2} \)
71 \( 1 - 3.25e5T + 1.28e11T^{2} \)
73 \( 1 + 1.07e5iT - 1.51e11T^{2} \)
79 \( 1 + 3.39e5T + 2.43e11T^{2} \)
83 \( 1 - 2.24e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.83e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.26e4iT - 8.32e11T^{2} \)
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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

−12.78124877968528462223052962702, −11.68360935822968870678063013730, −10.94590954882672162668522690766, −9.325553616501600985696200279340, −7.63283653899079796439415605824, −6.79234516241518337092175636030, −6.05634794964856201339693192248, −4.04100611062674829161185961032, −2.60613574951610796281916221591, −1.19271860961090183350302941125, 1.35727151488341893449551637675, 3.62542378903969993238997353671, 4.44716147124685369100637714148, 5.47590609639659484430349269416, 6.99040577939503031742706854868, 8.874218727059248834692764100682, 9.456317757361505549570622031089, 10.83351356419199107640277370868, 11.90046487659916191762047672773, 12.76254994728925279536858346856

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