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Properties

Label 2-98-7.4-c3-0-1
Degree $2$
Conductor $98$
Sign $-0.266 - 0.963i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 98,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.251040 + 0.329988i\)
\(L(\frac12)\) \(\approx\) \(0.251040 + 0.329988i\)
\(L(\frac{5}{2})\) 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

−13.21432350276452330374150881458, −12.67181885780791612825545727794, −11.62377122573260762154199460961, −10.42502912410371014268286723448, −9.623647697334305041733006581318, −8.294477818061506228848922127506, −7.35017591501444721326801329283, −5.06981349546936126657096608251, −4.23762159964574222724258355985, −2.01068374645846887997242350460, 0.26356982747853890064697396602, 3.01448533349171404466831915028, 4.98467690433499839281171381842, 6.64398851792384477131846210852, 7.19961503421966122236019862978, 8.517043091481294006301526623494, 9.831290085868944363262197337612, 10.96534257935399045752644406775, 11.94300775304491176090849216710, 13.27523785637484099221252221109

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