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Properties

Label 2-98-49.25-c1-0-0
Degree $2$
Conductor $98$
Sign $0.106 - 0.994i$
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  + (−0.955 + 0.294i)2-s + (1.04 + 2.67i)3-s + (0.826 − 0.563i)4-s + (0.926 + 0.139i)5-s + (−1.79 − 2.24i)6-s + (−2.29 − 1.30i)7-s + (−0.623 + 0.781i)8-s + (−3.85 + 3.57i)9-s + (−0.926 + 0.139i)10-s + (2.63 + 2.44i)11-s + (2.37 + 1.61i)12-s + (0.637 − 2.79i)13-s + (2.58 + 0.574i)14-s + (0.599 + 2.62i)15-s + (0.365 − 0.930i)16-s + (0.306 − 4.09i)17-s + ⋯
L(s)  = 1  + (−0.675 + 0.208i)2-s + (0.605 + 1.54i)3-s + (0.413 − 0.281i)4-s + (0.414 + 0.0624i)5-s + (−0.731 − 0.916i)6-s + (−0.868 − 0.495i)7-s + (−0.220 + 0.276i)8-s + (−1.28 + 1.19i)9-s + (−0.293 + 0.0441i)10-s + (0.795 + 0.738i)11-s + (0.685 + 0.467i)12-s + (0.176 − 0.774i)13-s + (0.690 + 0.153i)14-s + (0.154 + 0.677i)15-s + (0.0913 − 0.232i)16-s + (0.0744 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.106 - 0.994i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663604 + 0.596617i\)
\(L(\frac12)\) \(\approx\) \(0.663604 + 0.596617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.955 - 0.294i)T \)
7 \( 1 + (2.29 + 1.30i)T \)
good3 \( 1 + (-1.04 - 2.67i)T + (-2.19 + 2.04i)T^{2} \)
5 \( 1 + (-0.926 - 0.139i)T + (4.77 + 1.47i)T^{2} \)
11 \( 1 + (-2.63 - 2.44i)T + (0.822 + 10.9i)T^{2} \)
13 \( 1 + (-0.637 + 2.79i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.306 + 4.09i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (0.389 + 0.674i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.376 - 5.01i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (-9.62 + 4.63i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-4.29 + 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.57 + 4.48i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (3.59 - 4.50i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-3.89 - 4.88i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (5.43 - 1.67i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (5.88 - 4.00i)T + (19.3 - 49.3i)T^{2} \)
59 \( 1 + (8.32 - 1.25i)T + (56.3 - 17.3i)T^{2} \)
61 \( 1 + (2.24 + 1.52i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (2.31 - 4.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.96 + 1.42i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (12.9 + 4.00i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (-0.591 - 1.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.87 + 8.22i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.49 + 2.31i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−14.38472782126634137001387416542, −13.47924358800480418384708653097, −11.73477469506596280299574965973, −10.34944668881526693529030563779, −9.781401325769296102716515460103, −9.163418345172646275870475134289, −7.71757049847362612521395123234, −6.17339239415298591145987420767, −4.50448349211330909838265504926, −3.03341246019200304119041217408, 1.61619719304104849354304703503, 3.15394490023641285029675394651, 6.29220720528973574749527071225, 6.76910354157949869834152662640, 8.454660805156140107512463134729, 8.866034430678507417958493937241, 10.32251938049380396898337726585, 11.95718244475327862519197935624, 12.47560304691782197826073379377, 13.62440956382765872200952814927

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