Properties

Label 2-98-49.43-c1-0-1
Degree $2$
Conductor $98$
Sign $0.580 - 0.814i$
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  + (−0.623 + 0.781i)2-s + (1.93 + 0.930i)3-s + (−0.222 − 0.974i)4-s + (1.21 + 0.584i)5-s + (−1.93 + 0.930i)6-s + (−2.20 − 1.45i)7-s + (0.900 + 0.433i)8-s + (0.997 + 1.25i)9-s + (−1.21 + 0.584i)10-s + (0.840 − 1.05i)11-s + (0.477 − 2.09i)12-s + (−3.94 + 4.94i)13-s + (2.51 − 0.820i)14-s + (1.80 + 2.26i)15-s + (−0.900 + 0.433i)16-s + (0.988 − 4.32i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (1.11 + 0.537i)3-s + (−0.111 − 0.487i)4-s + (0.543 + 0.261i)5-s + (−0.788 + 0.379i)6-s + (−0.835 − 0.549i)7-s + (0.318 + 0.153i)8-s + (0.332 + 0.416i)9-s + (−0.384 + 0.184i)10-s + (0.253 − 0.317i)11-s + (0.137 − 0.603i)12-s + (−1.09 + 1.37i)13-s + (0.672 − 0.219i)14-s + (0.465 + 0.583i)15-s + (−0.225 + 0.108i)16-s + (0.239 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961307 + 0.495177i\)
\(L(\frac12)\) \(\approx\) \(0.961307 + 0.495177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (2.20 + 1.45i)T \)
good3 \( 1 + (-1.93 - 0.930i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-1.21 - 0.584i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-0.840 + 1.05i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.94 - 4.94i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.988 + 4.32i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 1.39T + 19T^{2} \)
23 \( 1 + (1.24 + 5.47i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (1.36 - 5.96i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 4.88T + 31T^{2} \)
37 \( 1 + (0.318 - 1.39i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-5.29 - 2.54i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (7.44 - 3.58i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-2.40 + 3.01i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.97 - 13.0i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-6.07 + 2.92i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.30 - 10.1i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + (1.29 + 5.67i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.79 + 6.00i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 5.78T + 79T^{2} \)
83 \( 1 + (-4.84 - 6.07i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.48 - 8.13i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 3.70T + 97T^{2} \)
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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

−14.12825699890754658154712223693, −13.68813235926521174607821820919, −11.97747209446188699313344755583, −10.27090665287392952027124842115, −9.580040549840625429780683250601, −8.862694889670864001305886437268, −7.38362558521408960230214756194, −6.36465265555885198331374506072, −4.43617663889249583597787427477, −2.75713318302190064810109713720, 2.12008558596671037549705297270, 3.35409058302023683128703445666, 5.66357723442203188788943741923, 7.41419528076821429631595281677, 8.355678812295586837608354705753, 9.503576755192557633344629972424, 10.09479984262723697042185416481, 11.90187575303607461982817522607, 12.90080550349082719232476432732, 13.39397090291580593168898068150

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