Properties

Label 2-98-49.2-c1-0-1
Degree $2$
Conductor $98$
Sign $0.331 - 0.943i$
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 + (−0.784 + 1.99i)3-s + (0.826 + 0.563i)4-s + (−2.07 + 0.313i)5-s + (−1.33 + 1.67i)6-s + (2.53 − 0.770i)7-s + (0.623 + 0.781i)8-s + (−1.18 − 1.09i)9-s + (−2.07 − 0.313i)10-s + (1.11 − 1.03i)11-s + (−1.77 + 1.20i)12-s + (−1.28 − 5.62i)13-s + (2.64 + 0.00943i)14-s + (1.00 − 4.40i)15-s + (0.365 + 0.930i)16-s + (0.428 + 5.72i)17-s + ⋯
L(s)  = 1  + (0.675 + 0.208i)2-s + (−0.452 + 1.15i)3-s + (0.413 + 0.281i)4-s + (−0.930 + 0.140i)5-s + (−0.546 + 0.685i)6-s + (0.956 − 0.291i)7-s + (0.220 + 0.276i)8-s + (−0.393 − 0.365i)9-s + (−0.657 − 0.0991i)10-s + (0.336 − 0.311i)11-s + (−0.512 + 0.349i)12-s + (−0.356 − 1.56i)13-s + (0.707 + 0.00252i)14-s + (0.259 − 1.13i)15-s + (0.0913 + 0.232i)16-s + (0.104 + 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980512 + 0.694394i\)
\(L(\frac12)\) \(\approx\) \(0.980512 + 0.694394i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 + 0.770i)T \)
good3 \( 1 + (0.784 - 1.99i)T + (-2.19 - 2.04i)T^{2} \)
5 \( 1 + (2.07 - 0.313i)T + (4.77 - 1.47i)T^{2} \)
11 \( 1 + (-1.11 + 1.03i)T + (0.822 - 10.9i)T^{2} \)
13 \( 1 + (1.28 + 5.62i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.428 - 5.72i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (-2.53 + 4.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.232 + 3.10i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (4.81 + 2.31i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-0.901 - 1.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.09 - 6.19i)T + (13.5 - 34.4i)T^{2} \)
41 \( 1 + (-1.63 - 2.04i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-1.98 + 2.49i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-4.24 - 1.30i)T + (38.8 + 26.4i)T^{2} \)
53 \( 1 + (0.646 + 0.440i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (5.69 + 0.858i)T + (56.3 + 17.3i)T^{2} \)
61 \( 1 + (-7.43 + 5.06i)T + (22.2 - 56.7i)T^{2} \)
67 \( 1 + (-5.16 - 8.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.34 - 4.01i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (9.55 - 2.94i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (-8.33 + 14.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.62 - 7.12i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (10.2 + 9.47i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 - 2.41T + 97T^{2} \)
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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

−14.47837027768890093720118722332, −13.08221958543372277234200671185, −11.83397426506868459419480989803, −10.99879018745927332023483024148, −10.27222658063900206022093802962, −8.441968333787864431757769863155, −7.42308230759208803682122612631, −5.61232546333030217389155113597, −4.58587009703603917962520712262, −3.53641506959019022111852942030, 1.77563267511548744561888221794, 4.10798366805725048205329450227, 5.47049026587954135587901525833, 7.03931196260037190569389510389, 7.64639422766733711191828985727, 9.309572587074991989801908483932, 11.25034069133028145681633644653, 11.93182605540297168696049831620, 12.23767144634255949191311771706, 13.75221013327322546526549470108

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