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Properties

Label 2-1098-183.101-c1-0-9
Degree $2$
Conductor $1098$
Sign $0.986 + 0.165i$
Analytic cond. $8.76757$
Root an. cond. $2.96100$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.0820 + 0.142i)5-s + (−0.594 − 2.22i)7-s + (−0.707 + 0.707i)8-s + (0.0424 − 0.158i)10-s + (−2.55 + 2.55i)11-s + (−0.513 + 0.888i)13-s + (1.14 + 1.99i)14-s + (0.500 − 0.866i)16-s + (2.00 + 0.535i)17-s + (3.04 − 1.75i)19-s + 0.164i·20-s + (1.80 − 3.12i)22-s + (1.10 + 1.10i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.0367 + 0.0635i)5-s + (−0.224 − 0.839i)7-s + (−0.249 + 0.249i)8-s + (0.0134 − 0.0501i)10-s + (−0.769 + 0.769i)11-s + (−0.142 + 0.246i)13-s + (0.307 + 0.532i)14-s + (0.125 − 0.216i)16-s + (0.485 + 0.129i)17-s + (0.697 − 0.402i)19-s + 0.0367i·20-s + (0.384 − 0.666i)22-s + (0.230 + 0.230i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1098\)    =    \(2 \cdot 3^{2} \cdot 61\)
Sign: $0.986 + 0.165i$
Analytic conductor: \(8.76757\)
Root analytic conductor: \(2.96100\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1098} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1098,\ (\ :1/2),\ 0.986 + 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070863644\)
\(L(\frac12)\) \(\approx\) \(1.070863644\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
61 \( 1 + (-7.79 - 0.469i)T \)
good5 \( 1 + (0.0820 - 0.142i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.594 + 2.22i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.55 - 2.55i)T - 11iT^{2} \)
13 \( 1 + (0.513 - 0.888i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.00 - 0.535i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.04 + 1.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.10 - 1.10i)T + 23iT^{2} \)
29 \( 1 + (-9.16 - 2.45i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.47 + 9.22i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.12 + 4.12i)T + 37iT^{2} \)
41 \( 1 + 8.62T + 41T^{2} \)
43 \( 1 + (-7.44 + 1.99i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-7.88 + 4.55i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.12 + 1.12i)T + 53iT^{2} \)
59 \( 1 + (1.19 + 4.45i)T + (-51.0 + 29.5i)T^{2} \)
67 \( 1 + (-4.98 + 1.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.33 - 0.893i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.59 + 7.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.11 - 15.3i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (7.68 + 4.43i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.82 - 3.82i)T - 89iT^{2} \)
97 \( 1 + (2.50 - 1.44i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.939281657788570714738679869211, −9.078644204456186211486184967480, −8.110653328505855733943873759906, −7.26994120419198203119415231818, −6.89994823079451597366672895161, −5.61599815479276285095369850671, −4.72081234050658647279876714523, −3.49889430437414575251587740230, −2.30066563645654147703124470657, −0.813348659036409677275626263172, 0.956498274801420065676567544051, 2.61434020251486783066623275892, 3.19868007699459553939871316434, 4.80732441185815898868704497663, 5.70640484241127817481067847492, 6.56304713866169459781763058774, 7.59762491210547666496325618456, 8.502184692101494203003470734001, 8.809920151234542546841188991422, 10.12779592997410067295454908522

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