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Properties

Label 2-1098-9.4-c1-0-15
Degree $2$
Conductor $1098$
Sign $0.376 - 0.926i$
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  + (−0.5 − 0.866i)2-s + (1.20 + 1.24i)3-s + (−0.499 + 0.866i)4-s + (−0.417 + 0.723i)5-s + (0.479 − 1.66i)6-s + (−0.864 − 1.49i)7-s + 0.999·8-s + (−0.110 + 2.99i)9-s + 0.835·10-s + (−0.501 − 0.869i)11-s + (−1.68 + 0.417i)12-s + (−0.749 + 1.29i)13-s + (−0.864 + 1.49i)14-s + (−1.40 + 0.348i)15-s + (−0.5 − 0.866i)16-s + 1.43·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.693 + 0.720i)3-s + (−0.249 + 0.433i)4-s + (−0.186 + 0.323i)5-s + (0.195 − 0.679i)6-s + (−0.326 − 0.565i)7-s + 0.353·8-s + (−0.0369 + 0.999i)9-s + 0.264·10-s + (−0.151 − 0.262i)11-s + (−0.485 + 0.120i)12-s + (−0.207 + 0.360i)13-s + (−0.231 + 0.400i)14-s + (−0.362 + 0.0900i)15-s + (−0.125 − 0.216i)16-s + 0.347·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1098\)    =    \(2 \cdot 3^{2} \cdot 61\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(8.76757\)
Root analytic conductor: \(2.96100\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1098} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1098,\ (\ :1/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.356078423\)
\(L(\frac12)\) \(\approx\) \(1.356078423\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.20 - 1.24i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.417 - 0.723i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.864 + 1.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.501 + 0.869i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.749 - 1.29i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 - 6.31T + 19T^{2} \)
23 \( 1 + (1.98 - 3.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.95 - 5.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.74 - 6.49i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.46T + 37T^{2} \)
41 \( 1 + (2.17 - 3.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.10 + 1.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.60 - 2.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.59T + 53T^{2} \)
59 \( 1 + (5.53 - 9.57i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (1.12 - 1.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 8.86T + 73T^{2} \)
79 \( 1 + (-3.34 - 5.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.98 + 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.769T + 89T^{2} \)
97 \( 1 + (-2.59 - 4.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.991398954078720736921003880215, −9.339830976154369176335334235338, −8.607133344507745464552520207364, −7.58706046576660631039906449815, −7.08920131098779524183609267033, −5.50796472067872352736264223437, −4.56682908242154961096991646109, −3.38939577712861788232868683189, −3.09736719651280003751729595233, −1.49307764228737981360906848148, 0.65193673965809840942270265794, 2.17617278088637581087391258131, 3.25689546362280700260598516869, 4.56649175068457705450726547949, 5.70863999653824271601032516812, 6.42185451994948234095814348700, 7.46428697921868152879914898031, 7.951927988398656018436301316348, 8.735565766596568658892654640657, 9.524544854097308015989169241718

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