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Properties

Label 2-98-49.29-c1-0-3
Degree $2$
Conductor $98$
Sign $0.541 + 0.840i$
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 + (0.0591 + 0.259i)3-s + (0.623 + 0.781i)4-s + (−0.871 − 3.81i)5-s + (0.0591 − 0.259i)6-s + (2.52 + 0.800i)7-s + (−0.222 − 0.974i)8-s + (2.63 − 1.27i)9-s + (−0.871 + 3.81i)10-s + (−1.24 − 0.601i)11-s + (−0.165 + 0.207i)12-s + (−0.394 − 0.189i)13-s + (−1.92 − 1.81i)14-s + (0.937 − 0.451i)15-s + (−0.222 + 0.974i)16-s + (−2.09 + 2.62i)17-s + ⋯
L(s)  = 1  + (−0.637 − 0.306i)2-s + (0.0341 + 0.149i)3-s + (0.311 + 0.390i)4-s + (−0.389 − 1.70i)5-s + (0.0241 − 0.105i)6-s + (0.953 + 0.302i)7-s + (−0.0786 − 0.344i)8-s + (0.879 − 0.423i)9-s + (−0.275 + 1.20i)10-s + (−0.376 − 0.181i)11-s + (−0.0478 + 0.0599i)12-s + (−0.109 − 0.0526i)13-s + (−0.514 − 0.485i)14-s + (0.242 − 0.116i)15-s + (−0.0556 + 0.243i)16-s + (−0.508 + 0.637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.541 + 0.840i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.696264 - 0.379974i\)
\(L(\frac12)\) \(\approx\) \(0.696264 - 0.379974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 - 0.800i)T \)
good3 \( 1 + (-0.0591 - 0.259i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.871 + 3.81i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.24 + 0.601i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.394 + 0.189i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.09 - 2.62i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 3.22T + 19T^{2} \)
23 \( 1 + (-5.72 - 7.18i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (2.68 - 3.36i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 - 4.95T + 31T^{2} \)
37 \( 1 + (2.75 - 3.45i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (2.64 + 11.5i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.674 + 2.95i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.899 - 0.433i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-5.43 - 6.81i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (2.32 - 10.1i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-4.57 + 5.74i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 - 3.79T + 67T^{2} \)
71 \( 1 + (5.19 + 6.51i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (7.97 - 3.84i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 + (10.1 - 4.87i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-3.75 + 1.80i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 3.08T + 97T^{2} \)
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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

−13.37795084206556843424716479823, −12.55370373944095469005773269103, −11.71623841394232769944052023590, −10.51075131273206518074471079812, −9.098762765131412559349458044832, −8.561861032451249241267765496770, −7.39704305542919732584222745394, −5.30092088495677862127681954800, −4.13384230557570174581570126113, −1.45343752279839804166114308586, 2.44136407474295386461113696168, 4.57138769848863469839456326235, 6.63366158254013822909221485719, 7.31286423857755941032462261239, 8.289224673464820539078093401323, 10.05706908463959819418665484951, 10.77572664984244673076275912871, 11.52732314555663865473598700103, 13.21685256782907137453924795664, 14.47552429420555759784849538393

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