Properties

Label 2-98-7.2-c1-0-2
Degree $2$
Conductor $98$
Sign $0.827 + 0.561i$
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  + (−0.5 + 0.866i)2-s + (−0.707 − 1.22i)3-s + (−0.499 − 0.866i)4-s + (1.41 − 2.44i)5-s + 1.41·6-s + 0.999·8-s + (0.500 − 0.866i)9-s + (1.41 + 2.44i)10-s + (1 + 1.73i)11-s + (−0.707 + 1.22i)12-s − 4·15-s + (−0.5 + 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.499 + 0.866i)18-s + (−3.53 + 6.12i)19-s − 2.82·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.408 − 0.707i)3-s + (−0.249 − 0.433i)4-s + (0.632 − 1.09i)5-s + 0.577·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.447 + 0.774i)10-s + (0.301 + 0.522i)11-s + (−0.204 + 0.353i)12-s − 1.03·15-s + (−0.125 + 0.216i)16-s + (−0.171 − 0.297i)17-s + (0.117 + 0.204i)18-s + (−0.811 + 1.40i)19-s − 0.632·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778377 - 0.239171i\)
\(L(\frac12)\) \(\approx\) \(0.778377 - 0.239171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.53 - 6.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-4.24 - 7.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.41 + 2.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (0.707 + 1.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−13.76193809812566124817678659413, −12.68605884014301529294020347923, −12.12565423005745550882922386484, −10.36797756766802892753769301082, −9.287691601674165385886858031133, −8.299776803105340755450334301620, −6.89861159000452862380207722778, −5.94916008639523159694127953550, −4.61592896308588504442376428583, −1.40536345694091675296508686664, 2.55959814979660381660161526943, 4.24208572076520311235406283664, 5.89273232430019475483528776608, 7.26568928480256498708708581767, 8.934524743459123306035708801029, 9.987839736584902093994878144698, 10.84562124592142148262248640429, 11.36040525061874285112173456885, 13.00642752127750995048478279999, 13.91972604691128846033682844961

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