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Properties

Label 2-98-49.29-c1-0-1
Degree $2$
Conductor $98$
Sign $0.470 - 0.882i$
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 + (0.646 + 2.83i)3-s + (0.623 + 0.781i)4-s + (−0.790 − 3.46i)5-s + (−0.646 + 2.83i)6-s + (−1.76 − 1.96i)7-s + (0.222 + 0.974i)8-s + (−4.89 + 2.35i)9-s + (0.790 − 3.46i)10-s + (3.20 + 1.54i)11-s + (−1.81 + 2.27i)12-s + (−1.94 − 0.936i)13-s + (−0.741 − 2.53i)14-s + (9.29 − 4.47i)15-s + (−0.222 + 0.974i)16-s + (0.710 − 0.890i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (0.373 + 1.63i)3-s + (0.311 + 0.390i)4-s + (−0.353 − 1.54i)5-s + (−0.263 + 1.15i)6-s + (−0.668 − 0.743i)7-s + (0.0786 + 0.344i)8-s + (−1.63 + 0.785i)9-s + (0.250 − 1.09i)10-s + (0.965 + 0.464i)11-s + (−0.522 + 0.655i)12-s + (−0.539 − 0.259i)13-s + (−0.198 − 0.678i)14-s + (2.40 − 1.15i)15-s + (−0.0556 + 0.243i)16-s + (0.172 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.470 - 0.882i$
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: \((2,\ 98,\ (\ :1/2),\ 0.470 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18475 + 0.710716i\)
\(L(\frac12)\) \(\approx\) \(1.18475 + 0.710716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.76 + 1.96i)T \)
good3 \( 1 + (-0.646 - 2.83i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.790 + 3.46i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-3.20 - 1.54i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (1.94 + 0.936i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.710 + 0.890i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 0.898T + 19T^{2} \)
23 \( 1 + (-1.90 - 2.38i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (3.37 - 4.23i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 + (-5.72 + 7.17i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-0.336 - 1.47i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.58 + 6.93i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.92 - 1.88i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.23 + 2.79i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (0.881 - 3.86i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (6.17 - 7.74i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 + (-8.69 - 10.9i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-10.1 + 4.90i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 2.31T + 79T^{2} \)
83 \( 1 + (-11.0 + 5.32i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-8.86 + 4.27i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + 2.24T + 97T^{2} \)
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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

−14.34043583042919451829388722878, −13.10282673238624791917522254589, −12.20183748285721255960412204351, −10.85317596502060002292485695769, −9.505546389990129709731861185707, −8.986112992100521726241009678666, −7.44641451692968700062760593425, −5.44859380723996141521553086356, −4.41531746971194418110547273679, −3.66958636185893887280245458487, 2.32027816670838974588742307886, 3.38909006298755443598378308056, 6.17997860860585504525374823465, 6.68008326277095108339335931010, 7.78588901481677685021483893955, 9.367549970757056783307084467139, 11.03017441385928300141085715829, 11.87998787804992248830264365207, 12.67679498030774462538411054440, 13.72751146928111204087730436036

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