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Properties

Label 2-98-49.43-c1-0-0
Degree $2$
Conductor $98$
Sign $0.847 - 0.531i$
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 + (−2.16 − 1.04i)3-s + (−0.222 − 0.974i)4-s + (3.56 + 1.71i)5-s + (2.16 − 1.04i)6-s + (1.88 + 1.85i)7-s + (0.900 + 0.433i)8-s + (1.72 + 2.15i)9-s + (−3.56 + 1.71i)10-s + (2.46 − 3.09i)11-s + (−0.534 + 2.34i)12-s + (−1.13 + 1.42i)13-s + (−2.62 + 0.316i)14-s + (−5.92 − 7.43i)15-s + (−0.900 + 0.433i)16-s + (−0.288 + 1.26i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (−1.24 − 0.601i)3-s + (−0.111 − 0.487i)4-s + (1.59 + 0.768i)5-s + (0.882 − 0.425i)6-s + (0.712 + 0.701i)7-s + (0.318 + 0.153i)8-s + (0.574 + 0.719i)9-s + (−1.12 + 0.543i)10-s + (0.744 − 0.933i)11-s + (−0.154 + 0.675i)12-s + (−0.314 + 0.394i)13-s + (−0.702 + 0.0845i)14-s + (−1.52 − 1.91i)15-s + (−0.225 + 0.108i)16-s + (−0.0700 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.847 - 0.531i$
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: \((2,\ 98,\ (\ :1/2),\ 0.847 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716726 + 0.206261i\)
\(L(\frac12)\) \(\approx\) \(0.716726 + 0.206261i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 - 1.85i)T \)
good3 \( 1 + (2.16 + 1.04i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-3.56 - 1.71i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-2.46 + 3.09i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.13 - 1.42i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.288 - 1.26i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 3.15T + 19T^{2} \)
23 \( 1 + (0.458 + 2.00i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-1.60 + 7.04i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 3.94T + 31T^{2} \)
37 \( 1 + (-0.329 + 1.44i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (8.00 + 3.85i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (5.80 - 2.79i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (4.08 - 5.12i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.27 + 9.95i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-12.1 + 5.86i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-1.26 + 5.52i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 0.964T + 67T^{2} \)
71 \( 1 + (-1.18 - 5.21i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (2.54 + 3.19i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + (-0.791 - 0.992i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-8.05 - 10.1i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−14.17032239649103443233611861999, −13.04538612471627116289136945297, −11.65743738509788356725383902721, −10.89711581578203496565809523944, −9.745467625851075977802494148464, −8.509467753395110066181973531352, −6.71208155389893971530445344883, −6.17646559703690731558191087902, −5.29101808952861789614051743081, −1.88697651237190817207268261959, 1.58342403993645834481925152922, 4.53240492768645408092253158396, 5.38164204443653679497247414700, 6.85464172571150317457520674085, 8.754043164634301601887060830505, 9.934255246191391126149924623991, 10.38450934648967706885238764720, 11.56957810111836012014670422687, 12.56301429869614903680587971169, 13.58810880356107482231498993731

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