↑

Properties

Label 2-98-49.9-c1-0-0
Degree $2$
Conductor $98$
Sign $0.104 - 0.994i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.733 − 0.680i)2-s + (−2.81 − 0.424i)3-s + (0.0747 + 0.997i)4-s + (0.904 + 2.30i)5-s + (1.77 + 2.22i)6-s + (−1.30 + 2.29i)7-s + (0.623 − 0.781i)8-s + (4.88 + 1.50i)9-s + (0.904 − 2.30i)10-s + (−1.71 + 0.528i)11-s + (0.212 − 2.83i)12-s + (−1.16 + 5.12i)13-s + (2.52 − 0.795i)14-s + (−1.56 − 6.87i)15-s + (−0.988 + 0.149i)16-s + (−5.15 − 3.51i)17-s + ⋯
L(s)  = 1  + (−0.518 − 0.480i)2-s + (−1.62 − 0.245i)3-s + (0.0373 + 0.498i)4-s + (0.404 + 1.03i)5-s + (0.724 + 0.908i)6-s + (−0.494 + 0.869i)7-s + (0.220 − 0.276i)8-s + (1.62 + 0.501i)9-s + (0.286 − 0.728i)10-s + (−0.517 + 0.159i)11-s + (0.0614 − 0.819i)12-s + (−0.324 + 1.42i)13-s + (0.674 − 0.212i)14-s + (−0.405 − 1.77i)15-s + (−0.247 + 0.0372i)16-s + (−1.24 − 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.104 - 0.994i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269041 + 0.242205i\)
\(L(\frac12)\) \(\approx\) \(0.269041 + 0.242205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (1.30 - 2.29i)T \)
good3 \( 1 + (2.81 + 0.424i)T + (2.86 + 0.884i)T^{2} \)
5 \( 1 + (-0.904 - 2.30i)T + (-3.66 + 3.40i)T^{2} \)
11 \( 1 + (1.71 - 0.528i)T + (9.08 - 6.19i)T^{2} \)
13 \( 1 + (1.16 - 5.12i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (5.15 + 3.51i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (0.588 - 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.87 + 1.96i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (-0.395 + 0.190i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-5.26 - 9.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0500 - 0.667i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-3.47 + 4.35i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-0.256 - 0.322i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (4.51 + 4.18i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (-0.935 - 12.4i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-2.14 + 5.47i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (0.314 - 4.20i)T + (-60.3 - 9.09i)T^{2} \)
67 \( 1 + (0.229 + 0.396i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.78 - 4.71i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-7.71 + 7.15i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (1.29 - 2.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.80 - 7.89i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (4.69 + 1.44i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.96523729080902987520662405003, −12.65852209548095092221237076781, −11.84023841308583811175524555626, −11.03775746089876716083523614232, −10.20166769688199585877195782975, −9.032656466679493529560493963854, −6.93904990815506697274981346988, −6.45233361069911631990512535261, −4.89347190740361084570648009204, −2.44113791447162738575537478226, 0.59793103291736296211181593991, 4.60318569468059179528868758002, 5.59838297478103755122325483771, 6.59362982819806014166687680791, 8.039416797417034736497676777545, 9.579043435207501801256012602218, 10.44811516587585152646607701347, 11.25647629868768117809457634517, 12.86427029474946293859730720227, 13.18060813995993238675155136693

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