Properties

Label 2-98-7.4-c3-0-3
Degree $2$
Conductor $98$
Sign $-0.605 - 0.795i$
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 + (−0.5 + 0.866i)3-s + (−1.99 + 3.46i)4-s + (3.5 + 6.06i)5-s − 1.99·6-s − 7.99·8-s + (13 + 22.5i)9-s + (−7 + 12.1i)10-s + (−17.5 + 30.3i)11-s + (−1.99 − 3.46i)12-s − 66·13-s − 7·15-s + (−8 − 13.8i)16-s + (29.5 − 51.0i)17-s + (−26 + 45.0i)18-s + (68.5 + 118. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.0962 + 0.166i)3-s + (−0.249 + 0.433i)4-s + (0.313 + 0.542i)5-s − 0.136·6-s − 0.353·8-s + (0.481 + 0.833i)9-s + (−0.221 + 0.383i)10-s + (−0.479 + 0.830i)11-s + (−0.0481 − 0.0833i)12-s − 1.40·13-s − 0.120·15-s + (−0.125 − 0.216i)16-s + (0.420 − 0.728i)17-s + (−0.340 + 0.589i)18-s + (0.827 + 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 98,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.729474 + 1.47152i\)
\(L(\frac12)\) \(\approx\) \(0.729474 + 1.47152i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-3.5 - 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (17.5 - 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 66T + 2.19e3T^{2} \)
17 \( 1 + (-29.5 + 51.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-68.5 - 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 106T + 2.43e4T^{2} \)
31 \( 1 + (-37.5 + 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 498T + 6.89e4T^{2} \)
43 \( 1 - 260T + 7.95e4T^{2} \)
47 \( 1 + (85.5 + 148. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-208.5 + 361. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (8.5 - 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-25.5 - 44.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (219.5 - 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 784T + 3.57e5T^{2} \)
73 \( 1 + (-147.5 + 255. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-247.5 - 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 932T + 5.71e5T^{2} \)
89 \( 1 + (436.5 + 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 290T + 9.12e5T^{2} \)
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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

−14.08871258689931858751094523211, −12.83754641785020293197941347850, −11.92991772655909124463845465725, −10.32716730388733783864823702041, −9.701020819243641939159994261177, −7.80806810345649549707131232111, −7.16358221752932503464067860583, −5.57830663988827454210390884204, −4.51651740539359137357503085798, −2.54156757359145915234473106195, 0.896018045867936731791782438673, 2.89144620221992818889475911496, 4.62028414642847223046156168397, 5.81549740700953971713696179583, 7.33146557393786375184961152936, 8.958782792089562890978329087004, 9.825213500095759776591501489114, 11.04352786611972326818903308025, 12.23304108328036510648978664059, 12.85164898889506516891526159190

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