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Properties

Label 2-98-49.22-c1-0-4
Degree $2$
Conductor $98$
Sign $0.936 + 0.350i$
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  + (0.900 − 0.433i)2-s + (−0.163 + 0.717i)3-s + (0.623 − 0.781i)4-s + (0.617 − 2.70i)5-s + (0.163 + 0.717i)6-s + (−0.959 + 2.46i)7-s + (0.222 − 0.974i)8-s + (2.21 + 1.06i)9-s + (−0.617 − 2.70i)10-s + (−2.49 + 1.20i)11-s + (0.458 + 0.575i)12-s + (−3.14 + 1.51i)13-s + (0.205 + 2.63i)14-s + (1.83 + 0.885i)15-s + (−0.222 − 0.974i)16-s + (−2.73 − 3.43i)17-s + ⋯
L(s)  = 1  + (0.637 − 0.306i)2-s + (−0.0945 + 0.414i)3-s + (0.311 − 0.390i)4-s + (0.276 − 1.20i)5-s + (0.0668 + 0.292i)6-s + (−0.362 + 0.931i)7-s + (0.0786 − 0.344i)8-s + (0.738 + 0.355i)9-s + (−0.195 − 0.855i)10-s + (−0.753 + 0.362i)11-s + (0.132 + 0.166i)12-s + (−0.872 + 0.419i)13-s + (0.0547 + 0.704i)14-s + (0.474 + 0.228i)15-s + (−0.0556 − 0.243i)16-s + (−0.664 − 0.832i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30921 - 0.236743i\)
\(L(\frac12)\) \(\approx\) \(1.30921 - 0.236743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.959 - 2.46i)T \)
good3 \( 1 + (0.163 - 0.717i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-0.617 + 2.70i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (2.49 - 1.20i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.14 - 1.51i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (2.73 + 3.43i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 4.53T + 19T^{2} \)
23 \( 1 + (-2.00 + 2.51i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-5.83 - 7.31i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 9.13T + 31T^{2} \)
37 \( 1 + (5.22 + 6.55i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-0.176 + 0.773i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.242 - 1.06i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-2.62 + 1.26i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-5.20 + 6.52i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.751 + 3.29i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-0.246 - 0.308i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + (-4.55 + 5.71i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-8.26 - 3.98i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 5.33T + 79T^{2} \)
83 \( 1 + (-9.31 - 4.48i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-1.49 - 0.722i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−13.59730636603722228979766138791, −12.65300542391992877195293183610, −12.20584429421159708034857099414, −10.64311492438726900757527752980, −9.615122468578788961516135885578, −8.634729966257838188181773888026, −6.82933556719967716149731057745, −5.14335539949786575051012799161, −4.63549445869333901081070180359, −2.35372656330066813349789211593, 2.76625144220257193616456005888, 4.34040171938406370752787535726, 6.26278713254810850724340825691, 6.91241443781304656391214420230, 7.987383942601676258922518932970, 10.11489168807002712150192072281, 10.65071242222166492273104115342, 12.11732027742696831411737640960, 13.23689336853851294227031332230, 13.76475739624809533297282914724

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