Properties

Label 2-98-49.39-c1-0-3
Degree $2$
Conductor $98$
Sign $0.997 + 0.0657i$
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 + (0.520 − 0.354i)3-s + (0.955 + 0.294i)4-s + (−0.0807 − 1.07i)5-s + (0.567 − 0.273i)6-s + (−2.64 − 0.115i)7-s + (0.900 + 0.433i)8-s + (−0.951 + 2.42i)9-s + (0.0807 − 1.07i)10-s + (0.249 + 0.634i)11-s + (0.601 − 0.185i)12-s + (0.00686 − 0.00860i)13-s + (−2.59 − 0.507i)14-s + (−0.424 − 0.532i)15-s + (0.826 + 0.563i)16-s + (0.289 + 0.268i)17-s + ⋯
L(s)  = 1  + (0.699 + 0.105i)2-s + (0.300 − 0.204i)3-s + (0.477 + 0.147i)4-s + (−0.0361 − 0.482i)5-s + (0.231 − 0.111i)6-s + (−0.999 − 0.0435i)7-s + (0.318 + 0.153i)8-s + (−0.317 + 0.807i)9-s + (0.0255 − 0.340i)10-s + (0.0751 + 0.191i)11-s + (0.173 − 0.0535i)12-s + (0.00190 − 0.00238i)13-s + (−0.693 − 0.135i)14-s + (−0.109 − 0.137i)15-s + (0.206 + 0.140i)16-s + (0.0701 + 0.0650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.997 + 0.0657i$
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: \((2,\ 98,\ (\ :1/2),\ 0.997 + 0.0657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42062 - 0.0467219i\)
\(L(\frac12)\) \(\approx\) \(1.42062 - 0.0467219i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.115i)T \)
good3 \( 1 + (-0.520 + 0.354i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.0807 + 1.07i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-0.249 - 0.634i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.00686 + 0.00860i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.289 - 0.268i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (3.12 + 5.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.89 - 4.54i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.591 + 2.59i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.87 + 4.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.53 - 0.472i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (-7.12 - 3.42i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-7.84 + 3.77i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (11.1 + 1.68i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (-7.53 - 2.32i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.377 - 5.03i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-5.40 + 1.66i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (3.11 - 5.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.529 + 2.31i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (2.11 - 0.318i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (5.18 + 8.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.01 - 2.52i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (5.92 - 15.0i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−13.59956350854744481622306909054, −13.14686886248549498230813011633, −12.09365132724344609420964647352, −10.87357029861521985932253547499, −9.544502376676157143200625612121, −8.272206254575584040568553648709, −7.02094960583293323715219211454, −5.73162837018705308514023199133, −4.29946235734248983835968853248, −2.61114141782389390206647964347, 2.90182941408672796812239823819, 4.00174598202453897788633043892, 5.95628011681733755838344724050, 6.77875681010080725463191866791, 8.475564106707112246541376005081, 9.793315598140520143373849693028, 10.72867891348130689004770675710, 12.13644541396085454813501925919, 12.77038355622508684152012946662, 14.20385678826480242403968687695

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