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Properties

Label 2-98-49.36-c1-0-1
Degree $2$
Conductor $98$
Sign $0.999 + 0.0115i$
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 + (0.724 + 0.908i)3-s + (−0.900 + 0.433i)4-s + (2.67 + 3.35i)5-s + (0.724 − 0.908i)6-s + (−2.50 − 0.848i)7-s + (0.623 + 0.781i)8-s + (0.367 − 1.60i)9-s + (2.67 − 3.35i)10-s + (−1.02 − 4.50i)11-s + (−1.04 − 0.504i)12-s + (0.400 + 1.75i)13-s + (−0.269 + 2.63i)14-s + (−1.11 + 4.86i)15-s + (0.623 − 0.781i)16-s + (−3.98 − 1.92i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (0.418 + 0.524i)3-s + (−0.450 + 0.216i)4-s + (1.19 + 1.50i)5-s + (0.295 − 0.370i)6-s + (−0.947 − 0.320i)7-s + (0.220 + 0.276i)8-s + (0.122 − 0.535i)9-s + (0.847 − 1.06i)10-s + (−0.310 − 1.35i)11-s + (−0.302 − 0.145i)12-s + (0.111 + 0.486i)13-s + (−0.0719 + 0.703i)14-s + (−0.286 + 1.25i)15-s + (0.155 − 0.195i)16-s + (−0.967 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $0.999 + 0.0115i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :1/2),\ 0.999 + 0.0115i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07842 - 0.00620564i\)
\(L(\frac12)\) \(\approx\) \(1.07842 - 0.00620564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.848i)T \)
good3 \( 1 + (-0.724 - 0.908i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.67 - 3.35i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (1.02 + 4.50i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.400 - 1.75i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (3.98 + 1.92i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 + (-0.506 + 0.243i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (3.74 + 1.80i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 3.98T + 31T^{2} \)
37 \( 1 + (-4.92 - 2.37i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (4.17 + 5.23i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (2.18 - 2.74i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.863 - 3.78i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-6.98 + 3.36i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (2.86 - 3.58i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-2.84 - 1.37i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + 7.10T + 67T^{2} \)
71 \( 1 + (-4.25 + 2.04i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.39 - 6.12i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 6.81T + 79T^{2} \)
83 \( 1 + (1.92 - 8.41i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.712 + 3.12i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 - 7.25T + 97T^{2} \)
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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

−13.75901170361921230945362981604, −13.23903264601222065904774148293, −11.42825183238101858506521040974, −10.57662496961472219756207666237, −9.717915530913888337502658920774, −9.015176169850108885311556351270, −6.98310177255750154494551237011, −5.95090652658990109016365860423, −3.61915382838095152992075560236, −2.71635044611318517500586803665, 1.98003956830233001687235643341, 4.78012061838275708505303618880, 5.83450042752879311492264245420, 7.19398320526201237949051223623, 8.500798026902465275180752682229, 9.355601652753202884739797348859, 10.19430700760220105597330651158, 12.52626035149501808195246230192, 13.05180884811545174572614791507, 13.58244002677454177358699001799

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