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Properties

Label 2-98-49.11-c1-0-2
Degree $2$
Conductor $98$
Sign $0.879 + 0.475i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.733 − 0.680i)2-s + (1.17 − 0.177i)3-s + (0.0747 − 0.997i)4-s + (−1.10 + 2.81i)5-s + (0.743 − 0.932i)6-s + (1.42 − 2.23i)7-s + (−0.623 − 0.781i)8-s + (−1.50 + 0.465i)9-s + (1.10 + 2.81i)10-s + (−4.49 − 1.38i)11-s + (−0.0891 − 1.18i)12-s + (0.134 + 0.588i)13-s + (−0.473 − 2.60i)14-s + (−0.801 + 3.51i)15-s + (−0.988 − 0.149i)16-s + (0.679 − 0.463i)17-s + ⋯
L(s)  = 1  + (0.518 − 0.480i)2-s + (0.680 − 0.102i)3-s + (0.0373 − 0.498i)4-s + (−0.493 + 1.25i)5-s + (0.303 − 0.380i)6-s + (0.538 − 0.842i)7-s + (−0.220 − 0.276i)8-s + (−0.502 + 0.155i)9-s + (0.348 + 0.889i)10-s + (−1.35 − 0.418i)11-s + (−0.0257 − 0.343i)12-s + (0.0372 + 0.163i)13-s + (−0.126 − 0.695i)14-s + (−0.206 + 0.906i)15-s + (−0.247 − 0.0372i)16-s + (0.164 − 0.112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34420 - 0.339716i\)
\(L(\frac12)\) \(\approx\) \(1.34420 - 0.339716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-1.42 + 2.23i)T \)
good3 \( 1 + (-1.17 + 0.177i)T + (2.86 - 0.884i)T^{2} \)
5 \( 1 + (1.10 - 2.81i)T + (-3.66 - 3.40i)T^{2} \)
11 \( 1 + (4.49 + 1.38i)T + (9.08 + 6.19i)T^{2} \)
13 \( 1 + (-0.134 - 0.588i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.679 + 0.463i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-1.57 - 2.72i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.58 - 4.49i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (7.23 + 3.48i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-3.51 + 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.241 + 3.22i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (-6.42 - 8.05i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-1.87 + 2.34i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.03 + 1.88i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-0.728 + 9.72i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (0.426 + 1.08i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (-0.325 - 4.34i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (2.94 - 5.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.58 + 3.17i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-2.97 - 2.75i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (0.0139 + 0.0241i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.75 - 12.0i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (17.2 - 5.33i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−13.82900158642810701782615538444, −13.14368867977424730012546149326, −11.28504773328565285575189123884, −11.06820139114468957134881554433, −9.790922660789672927901331787963, −8.023607222651130790609700635490, −7.30051405220397241698412236094, −5.52642710121840796611660772594, −3.72485177624024365346199073352, −2.64916073174171280202743463012, 2.82642383627699801617575763669, 4.70726517102288278266733869654, 5.50360703624892161458740356458, 7.54724669150124378303207594909, 8.508134592772756756029288245082, 9.096352341246488718022589604680, 11.03779337461283873327418287899, 12.34282384112105494624618161994, 12.88282773786160867972015578996, 14.09899271186770813009787640634

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