Properties

Label 2-98-7.2-c1-0-0
Degree $2$
Conductor $98$
Sign $-0.198 - 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 98,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557551 + 0.681479i\)
\(L(\frac12)\) \(\approx\) \(0.557551 + 0.681479i\)
\(L(\frac{3}{2})\) 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

−14.79873358887225634268505457331, −13.52015138597281091569651096517, −11.92178790968214890177382213507, −10.76769677914572029183622287480, −9.813535206047039858663936301779, −8.806492093525821167723727426256, −7.41183225848240975897507852246, −6.57471526883428119238872318887, −4.65071630495262576914634603754, −3.23598228469011589466332382812, 1.42533684749379634478021716895, 3.55355603051224765841058400538, 5.18977681863156718533409327610, 7.27840207632221511058928819666, 8.227231595240179265721530576344, 9.037326691025925374064102858460, 10.43001136471146669499907544972, 11.82127296370000774139181023977, 12.43186196573432285639667464336, 13.40343875377819809261329804461

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