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Properties

Label 2-98-49.23-c1-0-2
Degree $2$
Conductor $98$
Sign $0.612 + 0.790i$
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  + (0.0747 − 0.997i)2-s + (1.95 − 0.601i)3-s + (−0.988 − 0.149i)4-s + (0.958 + 0.889i)5-s + (−0.454 − 1.99i)6-s + (−2.43 + 1.02i)7-s + (−0.222 + 0.974i)8-s + (0.964 − 0.657i)9-s + (0.958 − 0.889i)10-s + (−2.30 − 1.57i)11-s + (−2.01 + 0.304i)12-s + (−0.810 + 0.390i)13-s + (0.843 + 2.50i)14-s + (2.40 + 1.15i)15-s + (0.955 + 0.294i)16-s + (0.719 − 1.83i)17-s + ⋯
L(s)  = 1  + (0.0528 − 0.705i)2-s + (1.12 − 0.347i)3-s + (−0.494 − 0.0745i)4-s + (0.428 + 0.397i)5-s + (−0.185 − 0.812i)6-s + (−0.921 + 0.388i)7-s + (−0.0786 + 0.344i)8-s + (0.321 − 0.219i)9-s + (0.302 − 0.281i)10-s + (−0.694 − 0.473i)11-s + (−0.582 + 0.0878i)12-s + (−0.224 + 0.108i)13-s + (0.225 + 0.670i)14-s + (0.620 + 0.298i)15-s + (0.238 + 0.0736i)16-s + (0.174 − 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13543 - 0.556175i\)
\(L(\frac12)\) \(\approx\) \(1.13543 - 0.556175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 + 0.997i)T \)
7 \( 1 + (2.43 - 1.02i)T \)
good3 \( 1 + (-1.95 + 0.601i)T + (2.47 - 1.68i)T^{2} \)
5 \( 1 + (-0.958 - 0.889i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (2.30 + 1.57i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (0.810 - 0.390i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-0.719 + 1.83i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (-0.387 + 0.670i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.28 - 5.82i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-6.52 - 8.18i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (4.21 + 7.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.65 + 0.551i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (1.26 - 5.55i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (2.00 + 8.79i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-0.907 + 12.1i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (-6.87 - 1.03i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (-3.85 + 3.57i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (12.1 - 1.83i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (-0.734 - 1.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.74 - 4.69i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.00237 - 0.0316i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (6.94 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.26 + 4.46i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-2.43 + 1.65i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 - 9.86T + 97T^{2} \)
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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

−13.56542654100203951933072929508, −13.03648283304788764267815955812, −11.75663885594855483568148975160, −10.40197486889344287259714107471, −9.436654633191023392513908393076, −8.519682148146022845120027846277, −7.15174290855096719889254692963, −5.52988871076165327509959531835, −3.33585878712522198921358511317, −2.44346085859508157698623224379, 2.96074271739291827723826869601, 4.50209321831877829616566175586, 6.09805512566990279835227889894, 7.49605537823183155598202755696, 8.604287389463686030641281597070, 9.540678383574186501015694779712, 10.35584555790047967036320506430, 12.50198067447417121603333079597, 13.27428570164243260633669487971, 14.16582538746063549218203534076

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