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Properties

Label 2-98-7.4-c3-0-4
Degree $2$
Conductor $98$
Sign $0.991 - 0.126i$
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 + (1 − 1.73i)3-s + (−1.99 + 3.46i)4-s + (6 + 10.3i)5-s − 3.99·6-s + 7.99·8-s + (11.5 + 19.9i)9-s + (12 − 20.7i)10-s + (−24 + 41.5i)11-s + (3.99 + 6.92i)12-s + 56·13-s + 24·15-s + (−8 − 13.8i)16-s + (57 − 98.7i)17-s + (23 − 39.8i)18-s + (−1 − 1.73i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.192 − 0.333i)3-s + (−0.249 + 0.433i)4-s + (0.536 + 0.929i)5-s − 0.272·6-s + 0.353·8-s + (0.425 + 0.737i)9-s + (0.379 − 0.657i)10-s + (−0.657 + 1.13i)11-s + (0.0962 + 0.166i)12-s + 1.19·13-s + 0.413·15-s + (−0.125 − 0.216i)16-s + (0.813 − 1.40i)17-s + (0.301 − 0.521i)18-s + (−0.0120 − 0.0209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
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: \((2,\ 98,\ (\ :3/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.48465 + 0.0942163i\)
\(L(\frac12)\) \(\approx\) \(1.48465 + 0.0942163i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-6 - 10.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (24 - 41.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 56T + 2.19e3T^{2} \)
17 \( 1 + (-57 + 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-60 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 54T + 2.43e4T^{2} \)
31 \( 1 + (118 - 204. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (73 + 126. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 126T + 6.89e4T^{2} \)
43 \( 1 + 376T + 7.95e4T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (87 - 150. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (69 - 119. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (190 + 329. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-242 + 419. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 576T + 3.57e5T^{2} \)
73 \( 1 + (-575 + 995. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (388 + 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 378T + 5.71e5T^{2} \)
89 \( 1 + (-195 - 337. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−13.44779623812582447742930829332, −12.43388070187998076446156057238, −11.07973383950148494762162354925, −10.32589401020255674270607193649, −9.340154020472940110806975944910, −7.78175582400896839882650385764, −6.93993864312313608032232048251, −5.11461118933648415097306758072, −3.13294608843190552641965629398, −1.78164535812667258158828769471, 1.08630948304263023849913396543, 3.76242010308913674379068395951, 5.42074390602952823928037517410, 6.35751161311983565853535862198, 8.202330154278258437081232592380, 8.827683652095119099116219878102, 9.934656354206426949715162666662, 11.00895022412397452990427924593, 12.72137853165263605218661264443, 13.37267652831105180251835574175

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