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Properties

Label 2-98-49.15-c1-0-0
Degree $2$
Conductor $98$
Sign $0.0766 - 0.997i$
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  + (0.222 − 0.974i)2-s + (−1.91 + 2.40i)3-s + (−0.900 − 0.433i)4-s + (−1.88 + 2.36i)5-s + (1.91 + 2.40i)6-s + (2.31 + 1.27i)7-s + (−0.623 + 0.781i)8-s + (−1.43 − 6.28i)9-s + (1.88 + 2.36i)10-s + (−0.673 + 2.94i)11-s + (2.76 − 1.33i)12-s + (0.532 − 2.33i)13-s + (1.75 − 1.97i)14-s + (−2.06 − 9.06i)15-s + (0.623 + 0.781i)16-s + (0.579 − 0.278i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (−1.10 + 1.38i)3-s + (−0.450 − 0.216i)4-s + (−0.843 + 1.05i)5-s + (0.782 + 0.981i)6-s + (0.876 + 0.482i)7-s + (−0.220 + 0.276i)8-s + (−0.478 − 2.09i)9-s + (0.596 + 0.747i)10-s + (−0.202 + 0.889i)11-s + (0.799 − 0.385i)12-s + (0.147 − 0.646i)13-s + (0.470 − 0.528i)14-s + (−0.534 − 2.34i)15-s + (0.155 + 0.195i)16-s + (0.140 − 0.0676i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0766 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0766 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.0766 - 0.997i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.0766 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.480376 + 0.444850i\)
\(L(\frac12)\) \(\approx\) \(0.480376 + 0.444850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-2.31 - 1.27i)T \)
good3 \( 1 + (1.91 - 2.40i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (1.88 - 2.36i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (0.673 - 2.94i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.532 + 2.33i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.579 + 0.278i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 - 0.629T + 19T^{2} \)
23 \( 1 + (-8.46 - 4.07i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (3.84 - 1.85i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 2.51T + 31T^{2} \)
37 \( 1 + (-5.78 + 2.78i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-1.29 + 1.61i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-0.845 - 1.06i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.235 - 1.03i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (4.22 + 2.03i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-1.96 - 2.45i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-4.86 + 2.34i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 6.32T + 67T^{2} \)
71 \( 1 + (12.7 + 6.14i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (0.925 + 4.05i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + (1.69 + 7.43i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.265 + 1.16i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−14.76171809222557822162422236868, −12.69838853366520342251915606384, −11.48113929128894206303058459897, −11.17124549851811315714649168907, −10.31846588904839028173717034870, −9.184000407174074085882775033520, −7.41097329173747574532659461812, −5.59029305009382866855833442248, −4.59735871929845732514896525500, −3.29475713103215309238213487811, 0.921376717676076397996487143760, 4.54354436622485235572764229510, 5.57800108674166866365364481898, 6.92513975164218718725234097551, 7.85725983152581860806679099354, 8.677326466380657081865128670194, 11.06736114444974437951409002542, 11.66335628649647394868428412606, 12.76886205378837615701945866214, 13.38072929486547008565673031475

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