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Properties

Label 2-98-7.4-c3-0-7
Degree $2$
Conductor $98$
Sign $-0.198 + 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0750511 - 0.0917328i\)
\(L(\frac12)\) \(\approx\) \(0.0750511 - 0.0917328i\)
\(L(\frac{5}{2})\) 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

−12.93654515551185337009414049631, −12.15278416276201060214663877244, −11.17656927868192593657787202822, −9.701053079281238344731863713559, −8.748889376088170182067053094379, −7.56303018381291545856212536387, −5.67248152748547664248865299410, −4.75231984050374488329652596511, −3.98747034651942221345603214594, −0.06325900694067852553135960071, 2.18806336988336164400032956730, 3.79113970056088417285437527670, 5.76271804229069354324198601339, 7.05480315751233636208392848719, 7.59455135522075174003361028642, 9.769315825724966092190208260970, 11.00429238135319684044819983176, 11.69373051309877324803011934381, 12.38314629547664370236912342926, 13.54500556386170296814692520670

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