Properties

Label 2-98-49.11-c3-0-8
Degree $2$
Conductor $98$
Sign $0.847 + 0.530i$
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  + (1.46 − 1.36i)2-s + (8.68 − 1.30i)3-s + (0.298 − 3.98i)4-s + (−3.71 + 9.46i)5-s + (10.9 − 13.7i)6-s + (15.2 + 10.4i)7-s + (−4.98 − 6.25i)8-s + (47.9 − 14.7i)9-s + (7.42 + 18.9i)10-s + (−17.7 − 5.47i)11-s + (−2.62 − 35.0i)12-s + (−10.3 − 45.4i)13-s + (36.6 − 5.44i)14-s + (−19.8 + 87.0i)15-s + (−15.8 − 2.38i)16-s + (−100. + 68.6i)17-s + ⋯
L(s)  = 1  + (0.518 − 0.480i)2-s + (1.67 − 0.251i)3-s + (0.0373 − 0.498i)4-s + (−0.332 + 0.846i)5-s + (0.745 − 0.934i)6-s + (0.825 + 0.564i)7-s + (−0.220 − 0.276i)8-s + (1.77 − 0.547i)9-s + (0.234 + 0.598i)10-s + (−0.486 − 0.149i)11-s + (−0.0631 − 0.842i)12-s + (−0.221 − 0.970i)13-s + (0.699 − 0.103i)14-s + (−0.342 + 1.49i)15-s + (−0.247 − 0.0372i)16-s + (−1.43 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.04948 - 0.876296i\)
\(L(\frac12)\) \(\approx\) \(3.04948 - 0.876296i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.46 + 1.36i)T \)
7 \( 1 + (-15.2 - 10.4i)T \)
good3 \( 1 + (-8.68 + 1.30i)T + (25.8 - 7.95i)T^{2} \)
5 \( 1 + (3.71 - 9.46i)T + (-91.6 - 85.0i)T^{2} \)
11 \( 1 + (17.7 + 5.47i)T + (1.09e3 + 749. i)T^{2} \)
13 \( 1 + (10.3 + 45.4i)T + (-1.97e3 + 953. i)T^{2} \)
17 \( 1 + (100. - 68.6i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (45.0 + 78.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (101. + 68.9i)T + (4.44e3 + 1.13e4i)T^{2} \)
29 \( 1 + (-226. - 108. i)T + (1.52e4 + 1.90e4i)T^{2} \)
31 \( 1 + (-52.3 + 90.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-12.0 - 160. i)T + (-5.00e4 + 7.54e3i)T^{2} \)
41 \( 1 + (26.1 + 32.7i)T + (-1.53e4 + 6.71e4i)T^{2} \)
43 \( 1 + (153. - 192. i)T + (-1.76e4 - 7.75e4i)T^{2} \)
47 \( 1 + (150. - 140. i)T + (7.75e3 - 1.03e5i)T^{2} \)
53 \( 1 + (-29.4 + 392. i)T + (-1.47e5 - 2.21e4i)T^{2} \)
59 \( 1 + (-165. - 422. i)T + (-1.50e5 + 1.39e5i)T^{2} \)
61 \( 1 + (-16.2 - 216. i)T + (-2.24e5 + 3.38e4i)T^{2} \)
67 \( 1 + (-393. + 682. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-931. + 448. i)T + (2.23e5 - 2.79e5i)T^{2} \)
73 \( 1 + (283. + 263. i)T + (2.90e4 + 3.87e5i)T^{2} \)
79 \( 1 + (537. + 930. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (215. - 944. i)T + (-5.15e5 - 2.48e5i)T^{2} \)
89 \( 1 + (1.37e3 - 425. i)T + (5.82e5 - 3.97e5i)T^{2} \)
97 \( 1 - 1.03e3T + 9.12e5T^{2} \)
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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

−13.42375994658354274141829820506, −12.60221844901904213499731079235, −11.16452878803876886858506125233, −10.23755265818702962075130105821, −8.691583633153228562723133609395, −8.029757834315263617209778465868, −6.59973043754181366709298404195, −4.56493090989668810412239287346, −3.05239743053364496861716834374, −2.19838953407190441663813879971, 2.18733568440490556829112352857, 4.05091359807047241950210102026, 4.73183774533207882650534046591, 7.04064276411111855347984678788, 8.159772148707891909402172713963, 8.693332119240823491999466601852, 9.983574264909082308467867839131, 11.64906159900648961774822954255, 12.88664183048999338043754466554, 13.95331591246358005227164921337

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