Properties

Label 2-98-49.39-c1-0-1
Degree $2$
Conductor $98$
Sign $0.637 - 0.770i$
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  + (0.988 + 0.149i)2-s + (−1.46 + 1.00i)3-s + (0.955 + 0.294i)4-s + (0.169 + 2.25i)5-s + (−1.60 + 0.771i)6-s + (2.40 − 1.10i)7-s + (0.900 + 0.433i)8-s + (0.0601 − 0.153i)9-s + (−0.169 + 2.25i)10-s + (−2.21 − 5.65i)11-s + (−1.69 + 0.524i)12-s + (0.113 − 0.142i)13-s + (2.54 − 0.729i)14-s + (−2.50 − 3.14i)15-s + (0.826 + 0.563i)16-s + (−3.61 − 3.35i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (−0.848 + 0.578i)3-s + (0.477 + 0.147i)4-s + (0.0755 + 1.00i)5-s + (−0.654 + 0.315i)6-s + (0.909 − 0.415i)7-s + (0.318 + 0.153i)8-s + (0.0200 − 0.0511i)9-s + (−0.0534 + 0.713i)10-s + (−0.669 − 1.70i)11-s + (−0.490 + 0.151i)12-s + (0.0315 − 0.0396i)13-s + (0.679 − 0.194i)14-s + (−0.647 − 0.812i)15-s + (0.206 + 0.140i)16-s + (−0.876 − 0.813i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08138 + 0.508663i\)
\(L(\frac12)\) \(\approx\) \(1.08138 + 0.508663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-2.40 + 1.10i)T \)
good3 \( 1 + (1.46 - 1.00i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.169 - 2.25i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (2.21 + 5.65i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.113 + 0.142i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (3.61 + 3.35i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-1.31 - 2.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.05 + 2.83i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (2.05 - 9.02i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.668 + 1.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.71 + 0.838i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (0.529 + 0.255i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (7.91 - 3.81i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (8.72 + 1.31i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-0.187 - 0.0577i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.873 - 11.6i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (0.523 - 0.161i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (3.83 - 6.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.01 - 4.43i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-7.77 + 1.17i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-0.263 - 0.455i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.1 + 13.9i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-1.42 + 3.62i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 4.21T + 97T^{2} \)
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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

−14.11360635546730158668836879959, −13.27638000860000586878224188370, −11.51248064958105090319687118299, −11.04929190661923302481650248339, −10.42016709869119614270032900064, −8.388841630709497045434015457256, −7.01278011114903077797152053243, −5.74614948603747473146251465931, −4.75874305409681743692944509502, −3.06158043789261951888781983266, 1.83527280729847770990177835442, 4.61379085755875160002566026777, 5.33306653761653061474000378989, 6.72460224850930630923053169942, 8.023709924547199456468190292995, 9.498101296121498290910802161293, 11.05639925946563022591718484559, 11.91286520296746140073740190026, 12.69526673280448668589458185251, 13.33323815707852881037197629323

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