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Properties

Label 2-98-7.5-c2-0-4
Degree $2$
Conductor $98$
Sign $0.0633 + 0.997i$
Analytic cond. $2.67030$
Root an. cond. $1.63410$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (3.62 − 2.09i)3-s + (−0.999 − 1.73i)4-s + (−2.74 − 1.58i)5-s − 5.91i·6-s − 2.82·8-s + (4.24 − 7.34i)9-s + (−3.87 + 2.23i)10-s + (6.62 + 11.4i)11-s + (−7.24 − 4.18i)12-s + 5.49i·13-s − 13.2·15-s + (−2.00 + 3.46i)16-s + (11.7 − 6.77i)17-s + (−6.00 − 10.3i)18-s + (0.621 + 0.358i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (1.20 − 0.696i)3-s + (−0.249 − 0.433i)4-s + (−0.548 − 0.316i)5-s − 0.985i·6-s − 0.353·8-s + (0.471 − 0.816i)9-s + (−0.387 + 0.223i)10-s + (0.601 + 1.04i)11-s + (−0.603 − 0.348i)12-s + 0.422i·13-s − 0.882·15-s + (−0.125 + 0.216i)16-s + (0.690 − 0.398i)17-s + (−0.333 − 0.577i)18-s + (0.0327 + 0.0188i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(2.67030\)
Root analytic conductor: \(1.63410\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45772 - 1.36814i\)
\(L(\frac12)\) \(\approx\) \(1.45772 - 1.36814i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
7 \( 1 \)
good3 \( 1 + (-3.62 + 2.09i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (2.74 + 1.58i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.62 - 11.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 5.49iT - 169T^{2} \)
17 \( 1 + (-11.7 + 6.77i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.621 - 0.358i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-1.13 + 1.96i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 20.4T + 841T^{2} \)
31 \( 1 + (21.3 - 12.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (32.4 - 56.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 21.0iT - 1.68e3T^{2} \)
43 \( 1 - 6.48T + 1.84e3T^{2} \)
47 \( 1 + (41.3 + 23.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (11.0 + 19.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-72.5 + 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (57.3 + 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (46.3 + 80.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (113. - 65.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-38.1 + 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 107. iT - 6.88e3T^{2} \)
89 \( 1 + (-145. - 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 25.5iT - 9.40e3T^{2} \)
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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

−13.41769637658096450309277608788, −12.38872915901341349964807026010, −11.74361454272207753183171526658, −10.05658622161794148473515178448, −8.999103561798808942099459505974, −7.953640889798224486903233493717, −6.79140965754722714968400936207, −4.68415662604099754773808647517, −3.28655698630612857261358113689, −1.71645028168781663764494516786, 3.17965751167893178565607529241, 3.99908205434079249642961652218, 5.77391738536912077315832754134, 7.40591811615045968164589285997, 8.397298115573614762390082374820, 9.248946217556932842823891177419, 10.60244328355917641234987643107, 11.91169611779045331184897273249, 13.29117952968161783144432160372, 14.28174588481237362243771575303

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